A124313 -id:A124313 - cp's OEIS frontend

A124312 Expansion of g.f. x^3*(1 - x)/(1 - x - x^2 - x^3 - x^4 - x^5).

0, 0, 1, 0, 1, 2, 4, 8, 15, 30, 59, 116, 228, 448, 881, 1732, 3405, 6694, 13160, 25872, 50863, 99994, 196583, 386472, 759784, 1493696, 2936529, 5773064, 11349545, 22312618, 43865452, 86237208, 169537887, 333302710, 655255875, 1288199132

Offset: 1

Artur Jasinski, Oct 25 2006

Second column of the n-th power of pentanacci matrix {{1,1,1,1,1},{1,0,0,0,0}, {0,1,0,0,0}, {0,0,1,0,0}, {0,0,0,1,0}} read from bottom to top gives 5 numbers starting from position n.

a(n+5) equals the number of n-length binary words avoiding runs of zeros of lengths 5i+4, (i=0,1,2,...). - Milan Janjic, Feb 26 2015

Robert Israel, Table of n, a(n) for n = 1..3407
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1).

Cf. A001591, A124311, A124313, A124314.

R:=PowerSeriesRing(Integers(), 50); [0,0] cat Coefficients(R!( x^3*(1-x)^2/(1-2*x+x^6) )); // G. C. Greubel, Aug 25 2023

f:= gfun:-rectoproc({a(n)+a(n+1)+a(n+2)+a(n+3)+a(n+4)-a(n+5), a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 0}, a(n), remember):
seq(f(n),n=1..30); # Robert Israel, Apr 13 2017

CoefficientList[Series[(x^3-x^4)/(1-x-x^2-x^3-x^4-x^5), {x,0,50}], x]

def A124312_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^2*(1-x)^2/(1-2*x+x^6) ).list()
A124312_list(50) # G. C. Greubel, Aug 25 2023

a(n) = A001591(n+1) - A001591(n). - Greg Dresden and Leo Zhang, Jun 20 2025

Edited by N. J. A. Sloane, Oct 29 2006, Jul 14 2007

Name corrected by Robert Israel, Apr 13 2017

A124314 Expansion of -1/(1 + x + x^2 + x^3 + x^4 - x^5).

Original entry on oeis.org

-1, 1, 0, 0, 0, -2, 3, -1, 0, 0, -4, 8, -5, 1, 0, -8, 20, -18, 7, -1, -16, 48, -56, 32, -9, -31, 112, -160, 120, -50, -53, 255, -432, 400, -220, -56, 563, -1119, 1232, -840, 108, 1182, -2801, 3583, -2912, 1056, 2256, -6784, 9967, -9407, 5024

Offset: 0

Artur Jasinski, Oct 25 2006

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

Cf. A124312, A124313.

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (-1+x)/(1-2*x^5+x^6) )); // G. C. Greubel, Aug 25 2023

CoefficientList[Series[1/(-1-x-x^2-x^3-x^4+x^5), {x,0,50}], x]
LinearRecurrence[{-1,-1,-1,-1,1}, {-1,1,0,0,0}, 60] (* G. C. Greubel, Aug 25 2023 *)

Vec(1/(-1-x-x^2-x^3-x^4+x^5)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

def A124314_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (-1+x)/(1-2*x^5+x^6) ).list()
A124314_list(60) # G. C. Greubel, Aug 25 2023

A251653 5-step Fibonacci sequence starting with 0,0,1,0,0.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 4, 7, 14, 28, 55, 108, 212, 417, 820, 1612, 3169, 6230, 12248, 24079, 47338, 93064, 182959, 359688, 707128, 1390177, 2733016, 5372968, 10562977, 20766266, 40825404, 80260631, 157788246, 310203524, 609844071, 1198921876, 2357018348, 4633776065, 9109763884, 17909324244, 35208804417

Offset: 0

Arie Bos, Dec 06 2014

Doubling the entries > 1 as 1, 2, 2, 4, 4, 7, 7, 14, 14, 28, 28, 55, 55,... (offset 0) gives Nyblom's palindromic binary strings having no 5-runs of 1's. - R. J. Mathar, Mar 28 2025

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. A. Nyblom, Counting Palindromic Binary Strings Without r-Runs of Ones, J. Int. Seq. 16 (2013) #13.8.7, P_5(n)
H. Prodinger, Counting Palindromes According to r-Runs of Ones Using Generating Functions, J. Int. Seq. 17 (2014) # 14.6.2, even length, r=4.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1).

Other 5-step sequences are A000322, A001591, A023424, A074048, A124312, A124313, A122997, A135055, A135056, A145029

(see www.jsoftware.com) First construct the generating matrix
1  1  1  1  1
1  2  2  2  2
2  3  4  4  4
4  6  7  8  8
8 12 14 15 16
Given that matrix one can produce the first 5*200 numbers by
, M(+/ . *)^:(i.250) 0 0 1 0 0x

LinearRecurrence[{1, 1, 1, 1, 1}, {0, 0, 1, 0, 0}, 100] (* G. C. Greubel, May 27 2016 *)

a(n+5) = a(n) + a(n+1) + a(n+2) + a(n+3) + a(n+4).

G.f.: x^2*(x^2 + x - 1)/(x^5 + x^4 + x^3 + x^2 + x - 1). - Chai Wah Wu, May 27 2016

cp's OEIS Frontend

A124312 Expansion of g.f. x^3*(1 - x)/(1 - x - x^2 - x^3 - x^4 - x^5).

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A124314 Expansion of -1/(1 + x + x^2 + x^3 + x^4 - x^5).

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Magma

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SageMath

A251653 5-step Fibonacci sequence starting with 0,0,1,0,0.

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