FIBONACCI

[Xlambda]

FIBONACCI !! recursive !! calculates the nth term of the famous Fibonacci sequence . This is more of a fun quest than a useful function. ( did this on a YT channel also, had fun with Rico about this).
This reflects how limited our computers are. You will be surprised. This time is not about recursion "iterations" limit, is about the limit of the number of calculations that a computer can perform.
Open a new workbook, define the lambda FIBONACCI. Very important, first, try values of n less then 20!! . Call FIBONACC(15) ...(19) ..etc. and increase it bit by bit.
For what number "n" do you think that your computer will start struggling giving you the answer?? Share with us here, if you please, how long it took your computer to calculate FIBONACCI(35)..(38) and (40). Eventually what type of computer you are using. Processor ,RAM etc. After all, 40 is a very small number.? Give it a try!!
Note: Fibonacci sequence is this one : 1,1,2,3,5,8,13,21,34,55,89,144....the value of n term is the sum of the previous 2 terms fib( n)=fib(n-1) + fib(n-2), and fib(1)=1,fib(2)=1
Fortunately , Fibonacci has also an explicit solution (non recursive), you can define another lambda to check the results FIB( n)=LAMBDA( n,LET(a,SQRT(5),x,((1+a)/2)^n,y,((1-a)/2)^n,1/a*(x-y)))

=LAMBDA(n,IF(n<3,1,FIBBONACCI(n-1)+FIBONACCI(n-2)))

LAMBDA 6.0.xlsx
	A	B	C	D	E	F	G	H
1	FIBONACCI		recursive	ckeck(non recursive)
2		n value	=FIBONACCI(B3)	=FIB(B3)	time(can use phone's stopwatch)
3		15	610	610	instant
4		18	2584	2584	instant
5		20	6765	6765	instant
6		30	832040	832040	1s
7		35	9227465	9227465	?s	share this
8		38	39088169	39088169	?s	values
9		40	102334155	102334155	?s	with us ?
10
FIBONACCI post

Cell Formulas
Range	Formula
C2:D2	C2	=FORMULATEXT(C3)
C3:C9	C3	=FIBONACCI(B3)
D3:D9	D3	=FIB(B3)

 

[Xlambda]

RRA(a) Reversed Recursive Adding !! recursive !!

=LAMBDA(a,
    LET(l,LEN(a),m,MAX(l),c,COLUMNS(a),r,SEQUENCE(m),s,SEQUENCE(,c+m-1)-r+1,
       b,MMULT(SEQUENCE(,m)^0,IFERROR((s>0)*(s<=c)*INDEX(MID(REPLACE(a,1,0,REPT(0,m-l)),r,1),m-r+1,s),0)),
       IF(AND(LEN(b)=1),FILTER(b,SEQUENCE(,COLUMNS(b))<=XMATCH(TRUE,b<>0,,-1)),RRA(b))
    )
)

The only modifications made to RA (replace one r to m-r+1)
...INDEX(MID(REPLACE(a,1,0,REPT(0,m-l)),r,1),r,s)…to ...INDEX(MID(REPLACE(a,1,0,REPT(0,m-l)),r,1),m-r+1,s)
and to last part:
instead of filtering out leading 0's we have to filter out trailing 0's
… SEQUENCE(,COLUMNS(b))>=XMATCH(TRUE,b<>0) .. to … SEQUENCE(,COLUMNS(b))<=XMATCH(TRUE,b<>0,,-1)

LARGE NUM FIBONACCI.xlsx
	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V	W
1	Part 3. RRA. Reverse Recursive Adding Concept. Adding reversed numbers aligned to the left.
2	No leading 0's are necessary and results will always be reversed.
3
4		same example used for RA												why results will stay reversed
5														5
6		0	0	9	8									8
7		8	9	8	9
8														=RRA(N5+8)
9		reversed or flipped array												3	1
10
11		8	9											8
12		9	8	9	8									3	1
13
14		17	17	9	8									=RRA(N11:O11+N12:O12)
15														1	2
16		inserting 0's
17	*	17	17	09	08									notice that even from early stages using RRA will keep the
18														results reversed and aligned to the left.
19		this time will extract last digits as first rows,
20		scnd digits to the left as scnd row+ offset to the right, etc.												see exact construction, next post
21		7	7	9	8
22			1	1	0	0
23
24		7	8	10	8	0
25
26		10 has 2 digits, loop the process from *
27		07	08	10	08	00
28
29		7	8	0	8	0
30			0	0	1	0	0
31
32		7	8	0	9	0	0
33
34		filtering out trailing 0's
35		7	8	0	9
36
37		7809 is reversed 9087
38
LNF 3

Cell Formulas
Range	Formula
N8,N14	N8	=FORMULATEXT(N9)
N9:O9	N9	=RRA(N5+8)
N12	N12	=RRA(N9)
B14:E14	B14	=B11:E11+B12:E12
N15:O15	N15	=RRA(N11:O11+N12:O12)
B24:F24	B24	=B21:F21+B22:F22
B32:G32	B32	=B29:G29+B30:G30
Dynamic array formulas.

 

[Xlambda]

LNF( n ) Large Number Fibonacci calls RRA.

=LAMBDA(n,
    LET(f,IF({1,0},1,0),
       r,REDUCE(0,SEQUENCE( n),LAMBDA(v,i,LET(s,SEQUENCE(i),x,IFNA(INDEX(v,i-2,),0),y,INDEX(v,i-1,),z,RRA(x+y),w,IF(COLUMNS(z)=1,INDEX(z,{1,2}),z),IF(s<3,f,IF(s=i,IFERROR(w,0),v))))),
       x,INDEX(r,n,),m,COLUMNS(x),
       CONCAT(INDEX(x,SEQUENCE(,m,m,-1)))
    )
)

RRA(a)

=LAMBDA(a,
    LET(l,LEN(a),m,MAX(l),c,COLUMNS(a),r,SEQUENCE(m),s,SEQUENCE(,c+m-1)-r+1,
       b,MMULT(SEQUENCE(,m)^0,IFERROR((s>0)*(s<=c)*INDEX(MID(REPLACE(a,1,0,REPT(0,m-l)),r,1),m-r+1,s),0)),
       IF(AND(LEN(b)=1),FILTER(b,SEQUENCE(,COLUMNS(b))<=XMATCH(TRUE,b<>0,,-1)),RRA(b))
    )
)

These are the only functions we need to calculate large Fibonacci numbers.
The cost we have to pay is in calculation time. A lot of rows add up to the total number of iterations, and this is slowing down the process.
If we are patient enough, the function is capable of calculating up to F 156787 (result will have 32767 digits) without running in any other type of limitation.
This is how array construction works, followed by some large numbers calculations.

LARGE NUM FIBONACCI.xlsx
	A	B	C	D	E	F	G	H	I	J	K	L	M	N
1	LNF Large Number Fibonacci. The function.
2
3		This is how LNF construction using RRA will look, first part
4	B7:	=LAMBDA(n,LET(f,IF({1,0},1,0),r,REDUCE(0,SEQUENCE(n),LAMBDA(v,i,LET(s,SEQUENCE(i),x,IFERROR(INDEX(v,i-2,),0), y,INDEX(v,i-1,),z,RRA(x+y),w,IF(COLUMNS(z)=1,INDEX(z,{1,2}),z),IF(s<3,f,IF(s=i,IFERROR(w,0),v))))),r))(21)
5
6
7		1	0	#N/A	#N/A	#N/A		LNF last part, extracting, reversing and concat n th row
8		1	0	#N/A	#N/A	#N/A		… x,INDEX(r,n,),m,COLUMNS(x),CONCAT(INDEX(x,SEQUENCE(,m,m,-1)))))
9		2	0	#N/A	#N/A	#N/A
10		3	0	#N/A	#N/A	#N/A		=LET(x,INDEX(B7#,21,),m,COLUMNS(x),CONCAT(INDEX(x,SEQUENCE(,m,m,-1))))
11		5	0	#N/A	#N/A	#N/A		10946
12		8	0	#N/A	#N/A	#N/A
13		3	1	#N/A	#N/A	#N/A		=LNF(21)
14		1	2	#N/A	#N/A	#N/A		10946
15		4	3	#N/A	#N/A	#N/A
16		5	5	#N/A	#N/A	#N/A
17		9	8	#N/A	#N/A	#N/A
18		4	4	1	#N/A	#N/A
19		3	3	2	#N/A	#N/A
20		7	7	3	#N/A	#N/A
21		0	1	6	#N/A	#N/A
22		7	8	9	#N/A	#N/A
23		7	9	5	1	#N/A
24		4	8	5	2	#N/A
25		1	8	1	4	#N/A
26		5	6	7	6	#N/A
27		6	4	9	0	1
28
LNF 4

Cell Formulas
Range	Formula
B7:F27	B7	=LAMBDA(n,LET(f,IF({1,0},1,0),r,REDUCE(0,SEQUENCE(n),LAMBDA(v,i,LET(s,SEQUENCE(i),x,IFNA(INDEX(v,i-2,),0),y,INDEX(v,i-1,),z,RRA(x+y),w,IF(COLUMNS(z)=1,INDEX(z,{1,2}),z),IF(s<3,f,IF(s=i,IFERROR(w,0),v))))),r))(21)
H10,H13	H10	=FORMULATEXT(H11)
H11	H11	=LET(x,INDEX(B7#,21,),m,COLUMNS(x),CONCAT(INDEX(x,SEQUENCE(,m,m,-1))))
H14	H14	=LNF(21)
Dynamic array formulas.

 

[Xlambda]

Calculation times.
Attached, screen capture from Wolfram Alpha F 5000 .

LARGE NUM FIBONACCI.xlsx
	A	B	C	D	E
1	Large Number Fibonacci. Calculation times.
2
3		calc time	char length	=LNF(1000)
4		9.69s	209	43466557686937456435688527675040625802564660517371780402481729089536555417949051890403879840079255169295922593080322634775209689623239873322471161642996440906533187938298969649928516003704476137795166849228875
5				=LNF(1222)
6		18.39s	256	1079969233398873236032844293301653524976362936180771647488810146258428573892502087348778468320557178381011519862132501167447528535147702705724587018435771357806213382154720957836431378302535456607039572026816018665428571946697730583021094317239872427815311
7				=LNF(1477)
8		32.61s	309	211475298697902185255785861961179135570552502746803252174956226558634024323947666637137823932524397611864671566211908330263377425204552074188208686993669123754004340250943108709212299180422293009765404930508242975773774612140021599477983006713536106549441161323499077298115887067363710153036315849480388057657
9				=LNF(2000)
10		1m19.62s	418	4224696333392304878706725602341482782579852840250681098010280137314308584370130707224123599639141511088446087538909603607640194711643596029271983312598737326253555802606991585915229492453904998722256795316982874482472992263901833716778060607011615497886719879858311468870876264597369086722884023654422295243347964480139515349562972087652656069529806499841977448720155612802665404554171717881930324025204312082516817125
11				=LNF(5000)
12		19m56.37s	1045	3878968454388325633701916308325905312082127714646245106160597214895550139044037097010822916462210669479293452858882973813483102008954982940361430156911478938364216563944106910214505634133706558656238254656700712525929903854933813928836378347518908762970712033337052923107693008518093849801803847813996748881765554653788291644268912980384613778969021502293082475666346224923071883324803280375039130352903304505842701147635242270210934637699104006714174883298422891491273104054328753298044273676822977244987749874555691907703880637046832794811358973739993110106219308149018570815397854379195305617510761053075688783766033667355445258844886241619210553457493675897849027988234351023599844663934853256411952221859563060475364645470760330902420806382584929156452876291575759142343809142302917491088984155209854432486594079793571316841692868039545309545388698114665082066862897420639323438488465240988742395873801976993820317174208932265468879364002630797780058759129671389634214252579116872755600360311370547754724604639987588046985178408674382863125
13
14				=LAMBDA(s,c,MID(s,SEQUENCE(ROUNDUP((LEN(s)/c),0),,1,c),c))(D12,100)
15				3878968454388325633701916308325905312082127714646245106160597214895550139044037097010822916462210669
16				4792934528588829738134831020089549829403614301569114789383642165639441069102145056341337065586562382
17				5465670071252592990385493381392883637834751890876297071203333705292310769300851809384980180384781399
18				6748881765554653788291644268912980384613778969021502293082475666346224923071883324803280375039130352
19				9033045058427011476352422702109346376991040067141748832984228914912731040543287532980442736768229772
20				4498774987455569190770388063704683279481135897373999311010621930814901857081539785437919530561751076
21				1053075688783766033667355445258844886241619210553457493675897849027988234351023599844663934853256411
22				9522218595630604753646454707603309024208063825849291564528762915757591423438091423029174910889841552
23				0985443248659407979357131684169286803954530954538869811466508206686289742063932343848846524098874239
24				5873801976993820317174208932265468879364002630797780058759129671389634214252579116872755600360311370
25				547754724604639987588046985178408674382863125
26
LNF 5

Cell Formulas
Range	Formula
C4,C12,C10,C8,C6	C4	=LEN(D4)
D3,D5,D7,D9,D11,D14	D3	=FORMULATEXT(D4)
D4	D4	=LNF(1000)
D6	D6	=LNF(1222)
D8	D8	=LNF(1477)
D10	D10	=LNF(2000)
D12	D12	=LNF(5000)
D15:D25	D15	=LAMBDA(s,c,MID(s,SEQUENCE(ROUNDUP((LEN(s)/c),0),,1,c),c))(D12,100)
Dynamic array formulas.

 

[Xlambda]

To prove that LNF is recursive efficient, let's calculate the frequency distributions of iterations calculated by RRA for every 2 rows, maximum number of iterations needed by 2 rows, total number of iterations for entire array.
For that I will use a tool function that counts the iterations. 0 iterations means that the exit condition is met after first calculation.
I_RRA(a,[i ]) where i argument counts total number of iterations after exiting condition is met

=LAMBDA(a,[i ],
     LET(l,LEN(a),m,MAX(l),c,COLUMNS(a),r,SEQUENCE(m),s,SEQUENCE(,c+m-1)-r+1,
        b,MMULT(SEQUENCE(,m)^0,IFERROR((s>0)*(s<=c)*INDEX(MID(REPLACE(a,1,0,REPT(0,m-l)),r,1),m-r+1,s),0)),
        IF(AND(LEN(b)=1),i,I_RRA(b,i+1))
    )
)

Also, to be much easier will have ARLNF Array Reversed Large Number Fibonacci, which will return entire array, ( LNF without last part of extracting last row, reversing and concat, but without the NA()'s )
ARLNF( n)

=LAMBDA(n,
    LET(f,IF({1,0},1,0),
      r,REDUCE(0,SEQUENCE(n),LAMBDA(v,i,LET(s,SEQUENCE(i),x,IFNA(INDEX(v,i-2,),0),y,INDEX(v,i-1,),z,RRA(x+y),w,IF(COLUMNS(z)=1,INDEX(z,{1,2}),z),IF(s<3,f,IF(s=i,IFERROR(w,0),v))))),
     IFNA(r,"")
    )
)

LARGE NUM FIBONACCI.xlsx
	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q
1	ARLNF Array Reversed Large Number Fibonacci
2
3	n	=ARLNF(70)
4	1	1	0
5	2	1	0
6	3	2	0
7	4	3	0
8	5	5	0
9	6	8	0
10	7	3	1
11	8	1	2
12	9	4	3
13	10	5	5
14	11	9	8
15	12	4	4	1
16	13	3	3	2
17	14	7	7	3
18	15	0	1	6
19	16	7	8	9
20	17	7	9	5	1
21	18	4	8	5	2
22	19	1	8	1	4
23	20	5	6	7	6
24	21	6	4	9	0	1
25	22	1	1	7	7	1
26	23	7	5	6	8	2
27	24	8	6	3	6	4
28	25	5	2	0	5	7
29	26	3	9	3	1	2	1
30	27	8	1	4	6	9	1
31	28	1	1	8	7	1	3
32	29	9	2	2	4	1	5
33	30	0	4	0	2	3	8
34	31	9	6	2	6	4	3	1
35	32	9	0	3	8	7	1	2
36	33	8	7	5	4	2	5	3
37	34	7	8	8	2	0	7	5
38	35	5	6	4	7	2	2	9
39	36	2	5	3	0	3	9	4	1
40	37	7	1	8	7	5	1	4	2
41	38	9	6	1	8	8	0	9	3
42	39	6	8	9	5	4	2	3	6
43	40	5	5	1	4	3	3	2	0	1
44	41	1	4	1	0	8	5	5	6	1
45	42	6	9	2	4	1	9	7	6	2
46	43	7	3	4	4	9	4	3	3	4
47	44	3	3	7	8	0	4	1	0	7
48	45	0	7	1	3	0	9	4	3	1	1
49	46	3	0	9	1	1	3	6	3	8	1
50	47	3	7	0	5	1	2	1	7	9	2
51	48	6	7	9	6	2	5	7	0	8	4
52	49	9	4	0	2	4	7	8	7	7	7
53	50	5	2	0	9	6	2	6	8	5	2	1
54	51	4	7	0	1	1	0	5	6	3	0	2
55	52	9	9	0	0	8	2	1	5	9	2	3
56	53	3	7	1	1	9	2	6	1	3	3	5
57	54	2	7	2	1	7	5	7	6	2	6	8
58	55	5	4	4	2	6	8	3	8	5	9	3	1
59	56	7	1	7	3	3	4	1	5	8	5	2	2
60	57	2	6	1	6	9	2	5	3	4	5	6	3
61	58	9	7	8	9	2	7	6	8	2	1	9	5
62	59	1	4	0	6	2	0	2	2	7	6	5	9
63	60	0	2	9	5	5	7	8	0	0	8	4	5	1
64	61	1	6	9	1	8	7	0	3	7	4	0	5	2
65	62	1	8	8	7	3	5	9	3	7	2	5	0	4
66	63	2	4	8	9	1	3	0	7	4	7	5	5	6
67	64	3	2	7	7	5	8	9	0	2	0	1	6	0	1
68	65	5	6	5	7	7	1	0	8	6	7	6	1	7	1
69	66	8	8	2	5	3	0	0	9	8	7	7	7	7	2
70	67	3	5	8	2	1	2	0	7	5	5	4	9	4	4
71	68	1	4	1	8	4	2	0	6	4	3	2	7	2	7
72	69	4	9	9	0	6	4	0	3	0	9	6	6	7	1	1
73	70	5	3	1	9	0	7	0	9	4	2	9	3	0	9	1
74
LNF 6

Cell Formulas
Range	Formula
B3	B3	=FORMULATEXT(B4)
A4:A73	A4	=SEQUENCE(70)
B4:P73	B4	=ARLNF(70)
Dynamic array formulas.

 

[Xlambda]

Iteration's frequency distribution for F 1000

LARGE NUM FIBONACCI.xlsx
	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V	W	X	Y	Z	AA	AB
1	Entire Fibbonacci array for n=		1000	=SEQUENCE(C1)
2				↓	=LET(a,F4:HF4,b,F5:HF5,y,FILTER(b,b<>""),z,INDEX(a,SEQUENCE(,COLUMNS(y))),x,IF(z="",0,z),I_RRA(x+y))
3	max nr.			n	↓	=ARLNF(C1)
4	iterations			1	iterations	1	0
5	btwn 2 rows			2	0	1	0
6	=MAX(E5:E1003)			3	0	2	0
7	5			4	0	3	0
8				5	0	5	0
9	total nr. of			6	0	8	0
10	iterations			7	0	3	1
11	entire array			8	0	1	2
12	=SUM(E4:E1003)			9	0	4	3
13	1289			10	0	5	5
14		itererations		11	0	9	8
15	frequency	frequency		12	0	4	4	1
16	bins	distribution		13	0	3	3	2
17	0	90		14	0	7	7	3
18	1	575		15	0	0	1	6
19	2	291		16	0	7	8	9
20	3	41		17	0	7	9	5	1
21	4	1		18	0	4	8	5	2
22	5	1		19	0	1	8	1	4
23		0		20	0	5	6	7	6
24				21	0	6	4	9	0	1
25	for what n we have max nr. iterations			22	0	1	1	7	7	1
26	=FILTER(D4#,ISNUMBER(XMATCH(E4:E1003,A7)))			23	0	7	5	6	8	2
27	899			24	1	8	6	3	6	4
28				25	0	5	2	0	5	7
29				26	0	3	9	3	1	2	1
30				27	0	8	1	4	6	9	1
31				28	0	1	1	8	7	1	3
32				29	0	9	2	2	4	1	5
33				30	0	0	4	0	2	3	8
34				31	0	9	6	2	6	4	3	1
35				32	0	9	0	3	8	7	1	2
36				33	1	8	7	5	4	2	5	3
37				34	0	7	8	8	2	0	7	5
38				35	1	5	6	4	7	2	2	9
39				36	0	2	5	3	0	3	9	4	1
40				37	0	7	1	8	7	5	1	4	2
41				38	0	9	6	1	8	8	0	9	3
42				39	1	6	8	9	5	4	2	3	6
43				40	1	5	5	1	4	3	3	2	0	1
44				41	0	1	4	1	0	8	5	5	6	1
45				42	0	6	9	2	4	1	9	7	6	2
46				43	1	7	3	4	4	9	4	3	3	4
47				44	1	3	3	7	8	0	4	1	0	7
48				45	0	0	7	1	3	0	9	4	3	1	1
49				46	0	3	0	9	1	1	3	6	3	8	1
50				47	0	3	7	0	5	1	2	1	7	9	2
51				48	1	6	7	9	6	2	5	7	0	8	4
52				49	1	9	4	0	2	4	7	8	7	7	7
53				50	1	5	2	0	9	6	2	6	8	5	2	1
54				51	0	4	7	0	1	1	0	5	6	3	0	2
55				52	0	9	9	0	0	8	2	1	5	9	2	3
56				53	0	3	7	1	1	9	2	6	1	3	3	5
57				54	0	2	7	2	1	7	5	7	6	2	6	8
58				55	0	5	4	4	2	6	8	3	8	5	9	3	1
59				56	0	7	1	7	3	3	4	1	5	8	5	2	2
60				57	0	2	6	1	6	9	2	5	3	4	5	6	3
LNF 7

Cell Formulas
Range	Formula
D1,E2	D1	=FORMULATEXT(D4)
F3,A26,A12,A6	F3	=FORMULATEXT(F4)
D4:D1003	D4	=SEQUENCE(C1)
F4:HF1003	F4	=ARLNF(C1)
A7	A7	=MAX(E5:E1003)
A13	A13	=SUM(E4:E1003)
A17:A22	A17	=SEQUENCE(A7+1)-1
B17:B23	B17	=FREQUENCY(E5:E1003,A17#)
A27	A27	=FILTER(D4#,ISNUMBER(XMATCH(E4:E1003,A7)))
E5:E60	E5	=LET(a,F4:HF4,b,F5:HF5,y,FILTER(b,b<>""),z,INDEX(a,SEQUENCE(,COLUMNS(y))),x,IF(z="",0,z),I_RRA(x+y))
Dynamic array formulas.

 

[Xlambda]

Iteration's frequency distribution for F 5000

LARGE NUM FIBONACCI.xlsx
	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V	W	X	Y	Z	AA	AB
1	Entire Fibbonacci array for n=		5000	=SEQUENCE(C1)
2				↓	=LET(a,F4:ANJ4,b,F5:ANJ5,y,FILTER(b,b<>""),z,INDEX(a,SEQUENCE(,COLUMNS(y))),x,IF(z="",0,z),I_RRA(x+y))
3	max nr.			n	↓	=ARLNF(C1)
4	iterations			1	iterations	1	0
5	btwn 2 rows			2	0	1	0
6	=MAX(E5:E5003)			3	0	2	0
7	6			4	0	3	0
8				5	0	5	0
9	total nr. of			6	0	8	0
10	iterations			7	0	3	1
11	entire array			8	0	1	2
12	=SUM(E4:E5003)			9	0	4	3
13	9935			10	0	5	5
14		itererations		11	0	9	8
15	frequency	frequency		12	0	4	4	1
16	bins	distribution		13	0	3	3	2
17	0	90		14	0	7	7	3
18	1	1008		15	0	0	1	6
19	2	2900		16	0	7	8	9
20	3	889		17	0	7	9	5	1
21	4	101		18	0	4	8	5	2
22	5	10		19	0	1	8	1	4
23	6	1		20	0	5	6	7	6
24		0		21	0	6	4	9	0	1
25	for what n we have max nr. iterations			22	0	1	1	7	7	1
26	=FILTER(D4#,ISNUMBER(XMATCH(E4:E5003,A7)))			23	0	7	5	6	8	2
27	2201			24	1	8	6	3	6	4
28				25	0	5	2	0	5	7
29				26	0	3	9	3	1	2	1
30				27	0	8	1	4	6	9	1
31				28	0	1	1	8	7	1	3
32				29	0	9	2	2	4	1	5
33				30	0	0	4	0	2	3	8
34				31	0	9	6	2	6	4	3	1
35				32	0	9	0	3	8	7	1	2
36				33	1	8	7	5	4	2	5	3
37				34	0	7	8	8	2	0	7	5
38				35	1	5	6	4	7	2	2	9
39				36	0	2	5	3	0	3	9	4	1
40				37	0	7	1	8	7	5	1	4	2
41				38	0	9	6	1	8	8	0	9	3
42				39	1	6	8	9	5	4	2	3	6
43				40	1	5	5	1	4	3	3	2	0	1
44				41	0	1	4	1	0	8	5	5	6	1
45				42	0	6	9	2	4	1	9	7	6	2
46				43	1	7	3	4	4	9	4	3	3	4
47				44	1	3	3	7	8	0	4	1	0	7
48				45	0	0	7	1	3	0	9	4	3	1	1
49				46	0	3	0	9	1	1	3	6	3	8	1
50				47	0	3	7	0	5	1	2	1	7	9	2
51				48	1	6	7	9	6	2	5	7	0	8	4
52				49	1	9	4	0	2	4	7	8	7	7	7
53				50	1	5	2	0	9	6	2	6	8	5	2	1
54				51	0	4	7	0	1	1	0	5	6	3	0	2
55				52	0	9	9	0	0	8	2	1	5	9	2	3
56				53	0	3	7	1	1	9	2	6	1	3	3	5
57				54	0	2	7	2	1	7	5	7	6	2	6	8
58				55	0	5	4	4	2	6	8	3	8	5	9	3	1
59				56	0	7	1	7	3	3	4	1	5	8	5	2	2
60				57	0	2	6	1	6	9	2	5	3	4	5	6	3
LNF 7

Cell Formulas
Range	Formula
D1,E2	D1	=FORMULATEXT(D4)
F3,A26,A12,A6	F3	=FORMULATEXT(F4)
D4:D5003	D4	=SEQUENCE(C1)
F4:ANJ5003	F4	=ARLNF(C1)
A7	A7	=MAX(E5:E5003)
A13	A13	=SUM(E4:E5003)
A17:A23	A17	=SEQUENCE(A7+1)-1
B17:B24	B17	=FREQUENCY(E5:E5003,A17#)
A27	A27	=FILTER(D4#,ISNUMBER(XMATCH(E4:E5003,A7)))
E5:E60	E5	=LET(a,F4:ANJ4,b,F5:ANJ5,y,FILTER(b,b<>""),z,INDEX(a,SEQUENCE(,COLUMNS(y))),x,IF(z="",0,z),I_RRA(x+y))
Dynamic array formulas.

 

[Xlambda]

LNF for n=10,000
Attached, screen capture from Wolfram Alpha F 10,000 . Fibonacci 10000 - Wolfram|Alpha

LARGE NUM FIBONACCI.xlsx
	A	B	C	D
1	Large Number Fibonacci n=10,000
2
3			char length	=LNF(10000)
4			2090	33644764876431783266621612005107543310302148460680063906564769974680081442166662368155595513633734025582065332680836159373734790483865268263040892463056431887354544369559827491606602099884183933864652731300088830269235673613135117579297437854413752130520504347701602264758318906527890855154366159582987279682987510631200575428783453215515103870818298969791613127856265033195487140214287532698187962046936097879900350962302291026368131493195275630227837628441540360584402572114334961180023091208287046088923962328835461505776583271252546093591128203925285393434620904245248929403901706233888991085841065183173360437470737908552631764325733993712871937587746897479926305837065742830161637408969178426378624212835258112820516370298089332099905707920064367426202389783111470054074998459250360633560933883831923386783056136435351892133279732908133732642652633989763922723407882928177953580570993691049175470808931841056146322338217465637321248226383092103297701648054726243842374862411453093812206564914032751086643394517512161526545361333111314042436854805106765843493523836959653428071768775328348234345557366719731392746273629108210679280784718035329131176778924659089938635459327894523777674406192240337638674004021330343297496902028328145933418826817683893072003634795623117103101291953169794607632737589253530772552375943788434504067715555779056450443016640119462580972216729758615026968443146952034614932291105970676243268515992834709891284706740862008587135016260312071903172086094081298321581077282076353186624611278245537208532365305775956430072517744315051539600905168603220349163222640885248852433158051534849622434848299380905070483482449327453732624567755879089187190803662058009594743150052402532709746995318770724376825907419939632265984147498193609285223945039707165443156421328157688908058783183404917434556270520223564846495196112460268313970975069382648706613264507665074611512677522748621598642530711298441182622661057163515069260029861704945425047491378115154139941550671256271197133252763631939606902895650288268608362241082050562430701794976171121233066073310059947366875
5
6				=LAMBDA(s,c,MID(s,SEQUENCE(ROUNDUP((LEN(s)/c),0),,1,c),c))(D4,120)
7				336447648764317832666216120051075433103021484606800639065647699746800814421666623681555955136337340255820653326808361593
8				737347904838652682630408924630564318873545443695598274916066020998841839338646527313000888302692356736131351175792974378
9				544137521305205043477016022647583189065278908551543661595829872796829875106312005754287834532155151038708182989697916131
10				278562650331954871402142875326981879620469360978799003509623022910263681314931952756302278376284415403605844025721143349
11				611800230912082870460889239623288354615057765832712525460935911282039252853934346209042452489294039017062338889910858410
12				651831733604374707379085526317643257339937128719375877468974799263058370657428301616374089691784263786242128352581128205
13				163702980893320999057079200643674262023897831114700540749984592503606335609338838319233867830561364353518921332797329081
14				337326426526339897639227234078829281779535805709936910491754708089318410561463223382174656373212482263830921032977016480
15				547262438423748624114530938122065649140327510866433945175121615265453613331113140424368548051067658434935238369596534280
16				717687753283482343455573667197313927462736291082106792807847180353291311767789246590899386354593278945237776744061922403
17				376386740040213303432974969020283281459334188268176838930720036347956231171031012919531697946076327375892535307725523759
18				437884345040677155557790564504430166401194625809722167297586150269684431469520346149322911059706762432685159928347098912
19				847067408620085871350162603120719031720860940812983215810772820763531866246112782455372085323653057759564300725177443150
20				515396009051686032203491632226408852488524331580515348496224348482993809050704834824493274537326245677558790891871908036
21				620580095947431500524025327097469953187707243768259074199396322659841474981936092852239450397071654431564213281576889080
22				587831834049174345562705202235648464951961124602683139709750693826487066132645076650746115126775227486215986425307112984
23				411826226610571635150692600298617049454250474913781151541399415506712562711971332527636319396069028956502882686083622410
24				82050562430701794976171121233066073310059947366875
25
LNF 9

Cell Formulas
Range	Formula
D3,D6	D3	=FORMULATEXT(D4)
C4	C4	=LEN(D4)
D4	D4	=LNF(10000)
D7:D24	D7	=LAMBDA(s,c,MID(s,SEQUENCE(ROUNDUP((LEN(s)/c),0),,1,c),c))(D4,120)
Dynamic array formulas.

 

[Xlambda]

Iteration's frequency distribution for F 10,000

LARGE NUM FIBONACCI.xlsx
	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P	Q	R	S	T	U	V	W	X	Y	Z	AA	AB
1	Entire Fibbonacci array for n=		10000	=SEQUENCE(C1)
2				↓	=LET(a,F4:CBO4,b,F5:CBO5,y,FILTER(b,b<>""),z,INDEX(a,SEQUENCE(,COLUMNS(y))),x,IF(z="",0,z),I_RRA(x+y))
3	max nr.			n	↓	=ARLNF(C1)
4	iterations			1	iterations	1	0
5	btwn 2 rows			2	0	1	0
6	=MAX(E5:E10003)			3	0	2	0
7	6			4	0	3	0
8				5	0	5	0
9	total nr. of			6	0	8	0
10	iterations			7	0	3	1
11	entire array			8	0	1	2
12	=SUM(E4:E10003)			9	0	4	3
13	22736			10	0	5	5
14		itererations		11	0	9	8
15	frequency	frequency		12	0	4	4	1
16	bins	distribution		13	0	3	3	2
17	0	90		14	0	7	7	3
18	1	1017		15	0	0	1	6
19	2	5416		16	0	7	8	9
20	3	3066		17	0	7	9	5	1
21	4	365		18	0	4	8	5	2
22	5	41		19	0	1	8	1	4
23	6	4		20	0	5	6	7	6
24		0		21	0	6	4	9	0	1
25	for what n we have max nr. iterations			22	0	1	1	7	7	1
26	=FILTER(D4#,ISNUMBER(XMATCH(E4:E10003,A7)))			23	0	7	5	6	8	2
27	2201			24	1	8	6	3	6	4
28	6299			25	0	5	2	0	5	7
29	8790			26	0	3	9	3	1	2	1
30	9780			27	0	8	1	4	6	9	1
31				28	0	1	1	8	7	1	3
32				29	0	9	2	2	4	1	5
33				30	0	0	4	0	2	3	8
34				31	0	9	6	2	6	4	3	1
35				32	0	9	0	3	8	7	1	2
36				33	1	8	7	5	4	2	5	3
37				34	0	7	8	8	2	0	7	5
38				35	1	5	6	4	7	2	2	9
39				36	0	2	5	3	0	3	9	4	1
40				37	0	7	1	8	7	5	1	4	2
41				38	0	9	6	1	8	8	0	9	3
42				39	1	6	8	9	5	4	2	3	6
43				40	1	5	5	1	4	3	3	2	0	1
44				41	0	1	4	1	0	8	5	5	6	1
45				42	0	6	9	2	4	1	9	7	6	2
46				43	1	7	3	4	4	9	4	3	3	4
47				44	1	3	3	7	8	0	4	1	0	7
48				45	0	0	7	1	3	0	9	4	3	1	1
49				46	0	3	0	9	1	1	3	6	3	8	1
50				47	0	3	7	0	5	1	2	1	7	9	2
51				48	1	6	7	9	6	2	5	7	0	8	4
52				49	1	9	4	0	2	4	7	8	7	7	7
53				50	1	5	2	0	9	6	2	6	8	5	2	1
54				51	0	4	7	0	1	1	0	5	6	3	0	2
55				52	0	9	9	0	0	8	2	1	5	9	2	3
56				53	0	3	7	1	1	9	2	6	1	3	3	5
57				54	0	2	7	2	1	7	5	7	6	2	6	8
58				55	0	5	4	4	2	6	8	3	8	5	9	3	1
59				56	0	7	1	7	3	3	4	1	5	8	5	2	2
60				57	0	2	6	1	6	9	2	5	3	4	5	6	3
LNF 8

Cell Formulas
Range	Formula
D1,E2	D1	=FORMULATEXT(D4)
F3,A26,A12,A6	F3	=FORMULATEXT(F4)
D4:D10003	D4	=SEQUENCE(C1)
F4:CBO10003	F4	=ARLNF(C1)
A7	A7	=MAX(E5:E10003)
A13	A13	=SUM(E4:E10003)
A17:A23	A17	=SEQUENCE(A7+1)-1
B17:B24	B17	=FREQUENCY(E5:E10003,A17#)
A27:A30	A27	=FILTER(D4#,ISNUMBER(XMATCH(E4:E10003,A7)))
E5:E60	E5	=LET(a,F4:CBO4,b,F5:CBO5,y,FILTER(b,b<>""),z,INDEX(a,SEQUENCE(,COLUMNS(y))),x,IF(z="",0,z),I_RRA(x+y))
Dynamic array formulas.

 

[Xlambda]

Function of "n" that estimates total number of iterations "ni" for calculating LNF(n )
ni(n )=5E-05*n^2+1.8851*n-676.31

LARGE NUM FIBONACCI.xlsx
	A	B	C	D	E	F	G	H	I	J	K	L
1		=SCAN(0,'LNF 8'!E5:E10003,LAMBDA(v,x,v+x))
2		iterations			=XLOOKUP(D6#,A4#,B4#)
3	n	running tot			↓	=0.00005*D6#^2+1.8851*D6#-676.31			=0.00005*H6#^2+1.8851*H6#-676.31
4	2	0			total	function's			function's
5	3	0		n	iterations	estimation		n	estimation
6	4	0		500	517	279		1000	1,259
7	5	0		1000	1289	1,259		2000	3,294
8	6	0		1500	2198	2,264		3000	5,429
9	7	0		2000	3168	3,294		4000	7,664
10	8	0		2500	4202	4,349		5000	9,999
11	9	0		3000	5294	5,429		6000	12,434
12	10	0		3500	6438	6,534		7000	14,969
13	11	0		4000	7567	7,664		8000	17,604
14	12	0		4500	8736	8,819		9000	20,340
15	13	0		5000	9935	9,999		10000	23,175
16	14	0		5500	11149	11,204		11000	26,110
17	15	0		6000	12388	12,434		12000	29,145
18	16	0		6500	13640	13,689		13000	32,280
19	17	0		7000	14896	14,969		14000	35,515
20	18	0		7500	16165	16,274		15000	38,850
21	19	0		8000	17472	17,604		16000	42,285
22	20	0		8500	18778	18,960		17000	45,820
23	21	0		9000	20092	20,340		18000	49,455
24	22	0		9500	21417	21,745		19000	53,191
25	23	0		10000	22736	23,175		20000	57,026
26	24	1						21000	60,961
27	25	1						22000	64,996
28	26	1						23000	69,131
29	27	1						24000	73,366
30	28	1						25000	77,701
31	29	1						26000	82,136
32	30	1						27000	86,671
33	31	1						28000	91,306
34	32	1						29000	96,042
35	33	2						30000	100,877
36	34	2						31000	105,812
37	35	3						32000	110,847
38	36	3						33000	115,982
39	37	3						34000	121,217
40	38	3						35000	126,552
41	39	4						36000	131,987
42	40	5						37000	137,522
43	41	5						38000	143,157
44	42	5						39000	148,893
45	43	6						40000	154,728
46	44	7						41000	160,663
47	45	7						42000	166,698
48	46	7						43000	172,833
49	47	7						44000	179,068
50	48	8						45000	185,403
51	49	9						46000	191,838
52	50	10						47000	198,373
53	51	10						48000	205,008
54	52	10						49000	211,744
55	53	10						50000	218,579
56	54	10						51000	225,514
57	55	10						52000	232,549
58	56	10						53000	239,684
59	57	10						54000	246,919
60	58	11						55000	254,254
61	59	12						56000	261,689
62	60	13						57000	269,224
63	61	13						58000	276,859
64	62	14						59000	284,595
65	63	15						60000	292,430
66	64	16
LNF 10

Cell Formulas
Range	Formula
B1,F3,I3	B1	=FORMULATEXT(B4)
E2	E2	=FORMULATEXT(E6)
A4:A10002	A4	=SEQUENCE(10000-1)+1
B4:B10002	B4	=SCAN(0,'LNF 8'!E5:E10003,LAMBDA(v,x,v+x))
D6:D25	D6	=SEQUENCE(20,,500,500)
E6:E25	E6	=XLOOKUP(D6#,A4#,B4#)
F6:F25	F6	=0.00005*D6#^2+1.8851*D6#-676.31
H6:H65	H6	=SEQUENCE(60,,1000,1000)
I6:I65	I6	=0.00005*H6#^2+1.8851*H6#-676.31
Dynamic array formulas.

 

[jdellasala]

Lots of chatter here, but no mention that the LAMBDA posted has a typo. Posted is
"=LAMBDA(n,IF(n<3,1,FIBBONACCI(n-1)+FIBONACCI(n-2)))", but for any value 3 or more it just generates a #NAME! error. Should be

=LAMBDA(n,IF(n<3,1,FIBONACCI(n-1)+FIBONACCI(n-2)))

Just sayin'.