A130893 -id:A130893 - cp's OEIS frontend

A111958 Lucas numbers (A000032) mod 8.

2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7, 2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7, 2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7, 2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7, 2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7, 2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7, 2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7, 2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7, 2, 1, 3, 4, 7, 3, 2, 5, 7

Offset: 0

N. J. A. Sloane, Nov 28 2005

nonn

This sequence has period-length 12.

G. C. Greubel, Table of n, a(n) for n = 0..9999
Paulo Ribenboim, FFF (Favorite Fibonacci Flowers), Fib. Quart. 43 (No. 1, 2005), 3-14.
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,-1,1,-1,1,-1,1).

Cf. A000032, A079344, A130893.

LinearRecurrence[{1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1}, {2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3}, 105] (* Ray Chandler, Aug 27 2015 *)
Mod[LucasL[Range[0, 99]], 8] (* Alonso del Arte, Dec 19 2015 *)

From G. C. Greubel, Feb 08 2016: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) - a(n-6) + a(n-7) - a(n-8) + a(n-9) - a(n-10) + a(n-11).
a(n+12) = a(n). (End)

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

A268613 Lucas numbers mod 20.

Original entry on oeis.org

2, 1, 3, 4, 7, 11, 18, 9, 7, 16, 3, 19, 2, 1, 3, 4, 7, 11, 18, 9, 7, 16, 3, 19, 2, 1, 3, 4, 7, 11, 18, 9, 7, 16, 3, 19, 2, 1, 3, 4, 7, 11, 18, 9, 7, 16, 3, 19, 2, 1, 3, 4, 7, 11, 18, 9, 7, 16, 3, 19, 2, 1, 3, 4, 7, 11, 18, 9, 7, 16, 3, 19, 2, 1, 3, 4, 7, 11, 18, 9, 7, 16, 3, 19, 2, 1, 3, 4, 7, 11, 18, 9, 7, 16, 3, 19, 2, 1, 3, 4, 7, 11

Offset: 0

G. C. Greubel, Feb 08 2016

Sequence of period 12.

This is one of four sequences that are period 12 of the form Lucas[n] Mod x, where x is 8,10,20,40.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A000032, A111958, A130893.

[Lucas(n) mod 20: n in [0..100]]; // Vincenzo Librandi, Feb 09 2016

Table[Mod[LucasL[n],20],{n,0,100}]
PadRight[{},120,{2,1,3,4,7,11,18,9,7,16,3,19}] (* Harvey P. Dale, Jul 06 2025 *)

a(n+12) = a(n).

A137976 Fibonacci numbers (A000045) not in A103311.

Original entry on oeis.org

3, 13, 34, 144, 377, 1597, 4181, 17711, 46368, 196418, 514229, 2178309, 5702887, 24157817

Offset: 1

Paul Curtz, May 01 2008

nonn

Sequence of last digits: a(2n-1) mod 10 = a(2n) mod 10 = A130893(n). - Paul Curtz, May 07 2008

Edited by N. J. A. Sloane, May 08 2008

A268615 Lucas numbers mod 40.

Original entry on oeis.org

2, 1, 3, 4, 7, 11, 18, 29, 7, 36, 3, 39, 2, 1, 3, 4, 7, 11, 18, 29, 7, 36, 3, 39, 2, 1, 3, 4, 7, 11, 18, 29, 7, 36, 3, 39, 2, 1, 3, 4, 7, 11, 18, 29, 7, 36, 3, 39, 2, 1, 3, 4, 7, 11, 18, 29, 7, 36, 3, 39, 2, 1, 3, 4, 7, 11, 18, 29, 7, 36, 3, 39, 2, 1, 3, 4, 7, 11, 18, 29, 7, 36, 3, 39, 2, 1, 3, 4, 7, 11, 18, 29, 7, 36, 3, 39, 2, 1, 3, 4, 7, 11

Offset: 0

G. C. Greubel, Feb 08 2016

This sequence is of period 12.

This is one of four sequences that are period 12 of the form Lucas[n] Mod x, where x is 8,10,20,40.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A000032, A111958, A130893, A268613.

[Lucas(n) mod 40: n in [0..100]]; // Vincenzo Librandi, Feb 09 2016

Table[Mod[LucasL[n],40], {n,0,100}]
PadRight[{},120,{2,1,3,4,7,11,18,29,7,36,3,39}] (* Harvey P. Dale, Feb 04 2019 *)

a(n+12) = a(n).

A341414 a(n) = (Fibonacci(n)*Lucas(n)) mod 10.

Original entry on oeis.org

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

Offset: 0

Jens Rasmussen, Feb 11 2021

Fibonacci starting with 0,1 and Lucas starting with 2,1.
Blocks of 30 numbers with 10 even and 20 uneven numbers.
Symmetric as a(7-i)=a(8+i) for i=1,2,...,6, and a(22-j)=a(23+j) for j=1..21.
Decimal expansion of 13801675776055042253380279/999000999000999000999000999. - Jianing Song, Apr 04 2021

			For n=5: a(5) = (Fibonacci(5)*Lucas(5)) mod 10 = (5*11) mod 10 = 55 mod 10 = 5.

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

Cf. A000032, A000045, A001906, A130893.

Bisection of A003893.

Table[Mod[Fibonacci@n*LucasL@n, 10], {n, 0, 100}] (* Giorgos Kalogeropoulos, Mar 31 2021 *)

a(n) = fibonacci(2*(n%30)) % 10 \\ Jianing Song, Apr 04 2021

a(n) = (Fibonacci(n)*Lucas(n)) mod 10 = Fibonacci(2*n) mod 10 using Binet's formula for Fibonacci and corresponding formula for Lucas.
a(n) = a(n-30).
a(n) = a(n-3) - a(n-6) + a(n-9) - a(n-12) + a(n-15) - a(n-18) + a(n-21) - a(n-24) + a(n-27).
a(n) = A003893(2*n).

A262951 a(1) = 1, a(2) = 3, a(3) = 4 and for n>=4, a(n) = (a(n-3)+a(n-2)+a(n-1)+k) mod 10 where k = a(n/6) if n is divisible by 6, else 0.

Original entry on oeis.org

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

Offset: 1

Michel Marcus, Oct 05 2015

nonn

This sequence is similar to A130893. Every term of index k is the sum of the 3 preceding terms modulo 10, except that for every sixth term the sum includes also the term of index k/6.

Lambert gave this sequence in Anlage zur Architectonic as a kind of early pseudorandom sequence. - Charles R Greathouse IV, Oct 05 2015

			a(6) = 4+8+5 = (17 + a(6/6)) mod 10 = (17 + 1) mod 10 = 8.

Maarten Bullynck, L’histoire de l’informatique et l’histoire des mathématiques : rencontres, opportunités et écueils, Images des Mathématiques, CNRS, 2015 (in French).
Johann Heinrich Lambert, Anlage zur Architectonic, oder Theorie des Einfachen und des Ersten in der philosophischen und mathematischen Erkenntniß, 1771.
Index entries for sequences related to pseudo-random numbers.

Cf. A130893.

lista(nn) = {va = vector(nn); va[1] = 1; va[2] = 3; va[3] = 4; for (k=4, nn, va[k] = va[k-3] + va[k-2] + va[k-1]; if (! (k % 6) && (k > 6), va[k] += va[k/6]); va[k] = va[k] % 10;); va;}

a(n) = (a(n-3) + a(n-2) + a(n-1)) mod 10 if n is not a multiple of 6.

a(n) = (a(n-3) + a(n-2) + a(n-1) + a(n/6)) mod 10 if n is a multiple of 6.

cp's OEIS Frontend

A111958 Lucas numbers (A000032) mod 8.

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

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A268613 Lucas numbers mod 20.

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Magma

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A137976 Fibonacci numbers (A000045) not in A103311.

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A268615 Lucas numbers mod 40.

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A341414 a(n) = (Fibonacci(n)*Lucas(n)) mod 10.

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A262951 a(1) = 1, a(2) = 3, a(3) = 4 and for n>=4, a(n) = (a(n-3)+a(n-2)+a(n-1)+k) mod 10 where k = a(n/6) if n is divisible by 6, else 0.

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