A124312 -id:A124312 - cp's OEIS frontend

A124313 Pentanacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5), starting 1,0,0,0,1.

1, 0, 0, 0, 1, 2, 3, 6, 12, 24, 47, 92, 181, 356, 700, 1376, 2705, 5318, 10455, 20554, 40408, 79440, 156175, 307032, 603609, 1186664, 2332920, 4586400, 9016625, 17726218, 34848827, 68510990, 134689060, 264791720, 520566815, 1023407412

Offset: 1

Artur Jasinski, Oct 25 2006

G. C. Greubel, Table of n, a(n) for n = 1..1000
I. Flores, k-Generalized Fibonacci numbers, Fib. Quart., 5 (1967), 258-266.
Eric Weisstein's World of Mathematics, Pentanacci Number
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1).

Cf. A001591, A124312, A124314.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(1-2*x+x^4)/(1-2*x+x^6) )); // G. C. Greubel, Aug 25 2023

f[n_]:= MatrixPower[{{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}}, n][[ 1, 4]]; Array[f, 50]
LinearRecurrence[{1,1,1,1,1}, {1,0,0,0,1}, 40] (* G. C. Greubel, Aug 25 2023 *)

def A124313_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-2*x+x^4)/(1-2*x+x^6) ).list()
A124313_list(50) # G. C. Greubel, Aug 25 2023

G.f.: x*(1-x-x^2-x^3)/(1-x-x^2-x^3-x^4-x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009; checked and corrected by R. J. Mathar, Sep 16 2009

Edited by Ralf Stephan, Oct 20 2013

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

A124313 Pentanacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5), starting 1,0,0,0,1.

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

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SageMath

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

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