A003893 -id:A003893 - cp's OEIS frontend

A072675 Integers m such that the last digit of Fibonacci(m) is 1.

1, 2, 8, 19, 22, 28, 41, 59, 61, 62, 68, 79, 82, 88, 101, 119, 121, 122, 128, 139, 142, 148, 161, 179, 181, 182, 188, 199, 202, 208, 221, 239, 241, 242, 248, 259, 262, 268, 281, 299, 301, 302, 308, 319, 322, 328, 341, 359, 361, 362, 368, 379, 382, 388, 401

Offset: 1

Benoit Cloitre, Aug 07 2002

			Fibonacci(28) = 317811, so 28 is a term.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Cf. A000045, A003893.

Position[Fibonacci[Range[500]],?(Mod[#,10]==1&)]//Flatten (* or *) LinearRecurrence[{1,0,0,0,0,0,0,1,-1},{1,2,8,19,22,28,41,59,61},70] (* _Harvey P. Dale, Sep 17 2018 *)

Sequence contains numbers of form (1+60k) (2+60k) (8+60k) (19+60k) (22+60k) (28+60k) (41+60k) (59+60k) k>=0.

G.f.: x*(x^8+18*x^7+13*x^6+6*x^5+3*x^4+11*x^3+6*x^2+x+1) / (x^9-x^8-x+1). - Colin Barker, Jun 16 2013

A072682 Numbers congruent to {3, 36, 54, 57} mod 60.

Original entry on oeis.org

3, 36, 54, 57, 63, 96, 114, 117, 123, 156, 174, 177, 183, 216, 234, 237, 243, 276, 294, 297, 303, 336, 354, 357, 363, 396, 414, 417, 423, 456, 474, 477, 483, 516, 534, 537, 543, 576, 594, 597, 603, 636, 654, 657, 663, 696, 714, 717, 723, 756, 774, 777, 783

Offset: 1

Benoit Cloitre, Aug 07 2002

Numbers n such that the last digit of F(n) is 2 where F(n) is the n-th Fibonacci number.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A000045, A002265, A003893.

[n: n in [0..800] | n mod 60 in [3, 36, 54, 57]];  // Bruno Berselli, Jun 14 2016

A072682:=n->15*n+3*(1+I)*((1-I)*I^(2*n)-(5+2*I)*I^(-n)+(2+5*I)*I^n)/4: seq(A072682(n), n=1..100); # Wesley Ivan Hurt, Jun 14 2016

Select[Range[800], MemberQ[{3,36,54,57}, Mod[#,60]]&] (* Harvey P. Dale, Apr 07 2013 *)

Sequence contains numbers of the form: 3+60k, 36+60k, 54+60k, 57+60k, k>=0.
G.f.: 3*x*(1 + 11*x + 6*x^2 + x^3 + x^4) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jun 14 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = 15*n + 3*(1+i)*((1-i)*i^(2*n) - (5+2*i)*i^(-n) + (2+5*i)*i^n)/4 where i=sqrt(-1). (End)

Simpler definition from Ralf Stephan, Jun 18 2005

A072702 Last digit of F(n) is 4 where F(n) is the n-th Fibonacci number.

Original entry on oeis.org

9, 12, 18, 51, 69, 72, 78, 111, 129, 132, 138, 171, 189, 192, 198, 231, 249, 252, 258, 291, 309, 312, 318, 351, 369, 372, 378, 411, 429, 432, 438, 471, 489, 492, 498, 531, 549, 552, 558, 591, 609, 612, 618, 651, 669, 672, 678, 711, 729, 732, 738, 771, 789

Offset: 1

Benoit Cloitre, Aug 07 2002

Sequence contains numbers of form (9+60k), (12+60k), (18+60k), (51+60k), with k>=0.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A000045, A003893.

With[{fibs=Fibonacci[Range[800]]},Flatten[Position[fibs,?(Last[ IntegerDigits[ #]]==4&)]]] (* _Harvey P. Dale, Sep 24 2012 *)
Position[Mod[Fibonacci[Range[800]],10],4]//Flatten (* Harvey P. Dale, Apr 09 2023 *)

a(n) = (-60 + 6*(-1)^n + (9+21*I)*(-I)^n + (9-I*21)*I^n + 60*n) / 4 \\ Colin Barker, Oct 16 2015

Vec(x*(9*x^4+33*x^3+6*x^2+3*x+9)/(x^5-x^4-x+1) + O(x^100)) \\ Colin Barker, Oct 16 2015

for(n=0, 1e3, if(fibonacci(n) % 10 == 4, print1(n", "))) \\ Altug Alkan, Oct 16 2015

G.f.: x*(9*x^4+33*x^3+6*x^2+3*x+9) / (x^5-x^4-x+1). - Colin Barker, Jun 16 2013

a(n) = (-60 + 6*(-1)^n + (9+21*i)*(-i)^n + (9-i*21)*i^n + 60*n) / 4 where i=sqrt(-1). - Colin Barker, Oct 16 2015

A072703 Indices of Fibonacci numbers whose last digit is 5.

Original entry on oeis.org

5, 10, 20, 25, 35, 40, 50, 55, 65, 70, 80, 85, 95, 100, 110, 115, 125, 130, 140, 145, 155, 160, 170, 175, 185, 190, 200, 205, 215, 220, 230, 235, 245, 250, 260, 265, 275, 280, 290, 295, 305, 310, 320, 325, 335, 340, 350, 355, 365, 370, 380, 385, 395, 400, 410

Offset: 1

Benoit Cloitre, Aug 07 2002

Sequence contains numbers of the forms 5 + 60*k, 10 + 60*k, 20 + 60*k, 25 + 60*k, 35 + 60*k, 40 + 60*k, 50 + 60*k, 55 + 60*k, where k>=0.

Numbers that are congruent to {5, 10} mod 15. - Amiram Eldar, Jan 01 2022, Nov 25 2024

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000045, A001651, A003893, A248897.

Flatten[Position[Fibonacci[Range[400]],?(Last[IntegerDigits[#]]==5&)]] (* or *) LinearRecurrence[{1,1,-1},{5,10,20},60] (* or *) Table[-(5/4) (3+(-1)^n-6 n),{n,60}] (* _Harvey P. Dale, May 15 2011 *)

a(n) = 15*(n-1)-a(n-1), with a(1) = 5. - Vincenzo Librandi, Aug 08 2010
From Harvey P. Dale, May 15 2011: (Start)
a(1) = 5, a(2) = 10, a(3) = 20, a(n) = a(n-1)+a(n-2)-a(n-3).
a(n) = -(5/4)*(3+(-1)^n-6*n). (End)
G.f.: 5*x*(x^2+x+1) / ((x-1)^2*(x+1)). - Colin Barker, Jun 16 2013
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(15*sqrt(3)) = A248897 / 10. - Amiram Eldar, Jan 01 2022
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = cos(Pi/10)*sec(Pi/6) = sqrt((5+sqrt(5))/6).
Product_{n>=1} (1 + (-1)^n/a(n)) = (2/sqrt(3))*cos(7*Pi/30). (End)
a(n) = 5 * A001651(n). - Alois P. Heinz, Nov 27 2024

Edited by M. F. Hasler, Oct 08 2014

A072710 Last digit of F(n) is 8 where F(n) is the n-th Fibonacci number.

Original entry on oeis.org

6, 24, 27, 33, 66, 84, 87, 93, 126, 144, 147, 153, 186, 204, 207, 213, 246, 264, 267, 273, 306, 324, 327, 333, 366, 384, 387, 393, 426, 444, 447, 453, 486, 504, 507, 513, 546, 564, 567, 573, 606, 624, 627, 633, 666, 684, 687, 693, 726, 744, 747, 753, 786

Offset: 1

Benoit Cloitre, Aug 07 2002

Sequence contains numbers of form (6+60k), (24+60k), (27+60k), (33+60k), with k>=0.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A000045, A003893.

a(n) = (-60 - 6*(-1)^n - (21-9*I)*(-I)^n - (21+9*I)*I^n + 60*n) / 4 \\ Colin Barker, Oct 16 2015

Vec(x*(27*x^4+6*x^3+3*x^2+18*x+6)/(x^5-x^4-x+1) + O(x^100)) \\ Colin Barker, Oct 16 2015

G.f.: x*(27*x^4+6*x^3+3*x^2+18*x+6) / (x^5-x^4-x+1). - Colin Barker, Jun 16 2013

a(n) = (-60 - 6*(-1)^n - (21-9*i)*(-i)^n - (21+9*i)*i^n + 60*n) / 4 where i=sqrt(-1). - Colin Barker, Oct 16 2015

A074867 a(n) = M(a(n-1)) + M(a(n-2)) where a(1)=a(2)=1 and M(k) is the product of the digits of k in base 10.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 11, 4, 5, 9, 14, 13, 7, 10, 7, 7, 14, 11, 5, 6, 11, 7, 8, 15, 13, 8, 11, 9, 10, 9, 9, 18, 17, 15, 12, 7, 9, 16, 15, 11, 6, 7, 13, 10, 3, 3, 6, 9, 15, 14, 9, 13, 12, 5, 7, 12, 9, 11, 10, 1, 1, 2, 3, 5, 8, 13, 11, 4, 5, 9, 14, 13, 7, 10, 7, 7, 14, 11, 5, 6, 11, 7, 8, 15, 13

Offset: 1

Felice Russo, Sep 11 2002

Periodic with least period 60. - Christopher N. Swanson (cswanson(AT)ashland.edu), Jul 22 2003
From Hieronymus Fischer, Jul 01 2007: (Start)
The digital product analog (in base 10) of the Fibonacci recurrence.
a(n) and Fib(n)=A000045(n) are congruent modulo 10 which implies that (a(n) mod 10) is equal to (Fib(n) mod 10) = A003893(n). Thus (a(n) mod 10) is periodic with the Pisano period A001175(10)=60.
a(n)==A131297(n) modulo 10 (A131297(n)=digital sum analog base 11 of the Fibonacci recurrence).
For general bases p>1, we have the inequality 1<=a(n)<=2p-2 (for n>0). Actually, a(n)<=18.
(End)

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000045, A010073, A010074, A010075, A010076, A131294, A131295, A131296, A131297, A131318, A131319, A131320.

nxt[{a_,b_}]:={b,Times@@IntegerDigits[a]+Times@@IntegerDigits[b]}; Transpose[ NestList[nxt,{1,1},90]][[1]] (* Harvey P. Dale, Feb 01 2015 *)

From Hieronymus Fischer, Jul 01 2007: (Start)
a(n) = a(n-1)+a(n-2)-10*(floor(a(n-1)/10)+floor(a(n-2)/10)). This is valid, since a(n)<100.
a(n) = ds_10(a(n-1))+ds_10(a(n-2))-(floor(a(n-1)/10)+floor(a(n-2)/10)) where ds_10(x) is the digital sum of x in base 10.
a(n) = (a(n-1)mod 10)+(a(n-2)mod 10) = A010879(a(n-1))+A010879(a(n-2)).
a(n) = A131297(n) if A131297(n)<=10.
a(n) = Fib(n)-10*sum{1A000045(n).
a(n) = A000045(n)-10*sum{1A000045(n-k+1)*A059995(a(k))}. (End)

More terms from Christopher N. Swanson (cswanson(AT)ashland.edu), Jul 22 2003

Definition adapted to offset by Georg Fischer, Jun 18 2021

A105472 Next-to-last digit of n-th Fibonacci number in decimal representation, a(n) = 0 for n <= 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 6, 4, 1, 5, 6, 2, 9, 1, 1, 2, 4, 6, 0, 7, 8, 6, 5, 1, 6, 8, 5, 4, 9, 3, 3, 7, 0, 7, 7, 4, 2, 7, 9, 7, 7, 4, 1, 6, 7, 4, 2, 6, 8, 4, 2, 6, 8, 5, 4, 9, 3, 2, 6, 9, 5, 5, 0, 5, 6, 2, 8, 0, 9, 9, 8, 8, 7, 5, 3, 8, 2, 0, 2, 3, 6, 0, 7, 7, 4, 2, 7, 0, 7, 7, 5

Offset: 0

Reinhard Zumkeller, Apr 09 2005

a(n) = floor(A105471(n)/10) = floor(A000045(n)/10) mod 10;
A105471(n) = a(n)*10 + A003893(n);
the sequence is periodic with period 300; all blocks of 300 successive terms contain 160 even and 140 odd numbers.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 601 terms from Alex Ermolaev)

Cf. A000045, A003893, A105471.

Array[Mod[Floor[Fibonacci[#]/10], 10] &, 105, 0] (* Michael De Vlieger, Jan 04 2018 *)
Join[{0,0,0,0,0,0,0},Table[IntegerDigits[Fibonacci[n]][[-2]],{n,7,120}]] (* Harvey P. Dale, Aug 22 2020 *)

a(n) = (fibonacci(n) % 100)\10; \\ Michel Marcus, Jan 05 2018

A072708 Last digit of F(n) is 6 where F(n) is the n-th Fibonacci number.

Original entry on oeis.org

21, 39, 42, 48, 81, 99, 102, 108, 141, 159, 162, 168, 201, 219, 222, 228, 261, 279, 282, 288, 321, 339, 342, 348, 381, 399, 402, 408, 441, 459, 462, 468, 501, 519, 522, 528, 561, 579, 582, 588, 621, 639, 642, 648, 681, 699, 702, 708, 741, 759, 762, 768, 801

Offset: 1

Benoit Cloitre, Aug 07 2002

Sequence contains numbers of form (21+60k), (39+60k), (42+60k), (48+60k), with k>=0.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A000045, A003893.

LinearRecurrence[{1,0,0,1,-1},{21,39,42,48,81},60] (* Harvey P. Dale, Aug 28 2017 *)

a(n) = (-3/4+(3*I)/4)*((1+I)*(-1)^n + (5+2*I)*(-I)^n + (2+5*I)*I^n - (10+10*I)*n) \\ Colin Barker, Oct 16 2015

Vec(x*(12*x^4+6*x^3+3*x^2+18*x+21)/(x^5-x^4-x+1) + O(x^100)) \\ Colin Barker, Oct 16 2015

G.f.: x*(12*x^4+6*x^3+3*x^2+18*x+21) / (x^5-x^4-x+1). - Colin Barker, Jun 16 2013

a(n) = (-3/4+(3*i)/4)*((1+i)*(-1)^n + (5+2*i)*(-i)^n + (2+5*i)*i^n - (10+10*i)*n) where i=sqrt(-1). - Colin Barker, Oct 16 2015

A141053 Most-significant decimal digit of Fibonacci(5n+3).

Original entry on oeis.org

2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 7, 8, 8, 9, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2

Offset: 0

Paul Curtz, Aug 01 2008

Leading digit of A134490(n).
From Johannes W. Meijer, Jul 06 2011: (Start)
The leading digit d, 1 <= d <= 9, of A141053 follows Benford’s Law. This law states that the probability for the leading digit is p(d) = log_10(1+1/d), see the examples.
We observe that the last digit of A134490(n), i.e. F(5*n+3) mod 10, leads to the Lucas sequence A000032(n) (mod 10), i.e. a repetitive sequence of 12 digits [2, 1, 3, 4, 7, 1, 8, 9, 7, 6, 3, 9] with p(0) = p(5) = 0, p(1) = p(3) = p(7) = p(9) = 1/6 and p(2) = p(4) = p(6) = p(8) = 1/12. This does not obey Benford’s Law, which would predict that the last digit would satisfy p(d) = 1/10, see the links. (End)

			From _Johannes W. Meijer_, Jul 06 2011: (Start)
d     p(N=2000) p(N=4000) p(N=6000) p(Benford)
1      0.29900   0.29950   0.30033   0.30103
2      0.17700   0.17675   0.17650   0.17609
3      0.12550   0.12525   0.12517   0.12494
4      0.09650   0.09675   0.09700   0.09691
5      0.07950   0.07950   0.07933   0.07918
6      0.06700   0.06675   0.06700   0.06695
7      0.05800   0.05825   0.05800   0.05799
8      0.05150   0.05125   0.05100   0.05115
9      0.04600   0.04600   0.04567   0.04576
Total  1.00000   1.00000   1.00000   1.00000 (End)

Hans J. H. Tuenter, Table of n, a(n) for n = 0..10000
Kevin Brown, Benford's Law.
Eric Weisstein's World of Mathematics, Benford's Law.
Wikipedia, Benford's Law.
Index entries for sequences related to Benford's law

Cf. A000045 (F(n)), A008963 (Initial digit F(n)), A105511-A105519, A003893 (F(n) mod 10), A130893, A186190 (First digit tribonacci), A008952 (Leading digit 2^n), A008905 (Leading digit n!), A045510, A112420 (Leading digit Collatz 3*n+1 starting with 1117065), A007524 (log_10(2)), A104140 (1-log_10(9)). - Johannes W. Meijer, Jul 06 2011

Cf. A001622, A097348.

A134490 := proc(n) combinat[fibonacci](5*n+3) ; end proc:
A141053 := proc(n) convert(A134490(n),base,10) ; op(-1,%) ; end proc:
seq(A141053(n),n=0..70) ; # R. J. Mathar, Jul 04 2011

Table[IntegerDigits[Fibonacci[5n+3]][[1]],{n,0,70}] (* Harvey P. Dale, Jun 22 2025 *)

a(n) = floor(F(5*n+3)/10^(floor(log(F(5*n+3))/log(10)))). - Johannes W. Meijer, Jul 06 2011

For n>0, a(n) = floor(10^{alpha*n+beta}), where alpha=5*log_10(phi)-1, beta=log_10(1+2/sqrt(5)), {x}=x-floor(x) denotes the fractional part of x, log_10(phi) = A097348, and phi = (1+sqrt(5))/2 = A001622. - Hans J. H. Tuenter, Aug 27 2025

Edited by Johannes W. Meijer, Jul 06 2011

A072683 Numbers k such that the last digit of F(k) is 3 where F(k) is the k-th Fibonacci number.

Original entry on oeis.org

4, 7, 13, 26, 44, 46, 47, 53, 64, 67, 73, 86, 104, 106, 107, 113, 124, 127, 133, 146, 164, 166, 167, 173, 184, 187, 193, 206, 224, 226, 227, 233, 244, 247, 253, 266, 284, 286, 287, 293, 304, 307, 313, 326, 344, 346, 347, 353, 364, 367, 373, 386, 404, 406, 407

Offset: 1

Benoit Cloitre, Aug 07 2002

Numbers that are congruent to {4, 7, 13, 26, 44, 46, 47, 53} mod 60. - Amiram Eldar, Jan 01 2022

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Cf. A000045, A003893.

Select[Range[500],Mod[Fibonacci[#],10]==3&] (* Harvey P. Dale, Jan 24 2012 *)
LinearRecurrence[{1,0,0,0,0,0,0,1,-1},{4,7,13,26,44,46,47,53,64},60] (* Harvey P. Dale, Apr 10 2022 *)

G.f.: x*(7*x^8+6*x^7+x^6+2*x^5+18*x^4+13*x^3+6*x^2+3*x+4) / ((1-x)^2*(1+x)*(1+x^2)*(1+x^4)). - Colin Barker, Jun 16 2013

a(n) ~ 15*n/2. - Stefano Spezia, Jul 09 2023

cp's OEIS Frontend

A072675 Integers m such that the last digit of Fibonacci(m) is 1.

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A072682 Numbers congruent to {3, 36, 54, 57} mod 60.

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A072702 Last digit of F(n) is 4 where F(n) is the n-th Fibonacci number.

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A072703 Indices of Fibonacci numbers whose last digit is 5.

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A072710 Last digit of F(n) is 8 where F(n) is the n-th Fibonacci number.

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A074867 a(n) = M(a(n-1)) + M(a(n-2)) where a(1)=a(2)=1 and M(k) is the product of the digits of k in base 10.

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A105472 Next-to-last digit of n-th Fibonacci number in decimal representation, a(n) = 0 for n <= 6.

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