A135055 -id:A135055 - cp's OEIS frontend

A135056 Pentanacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) if n>=5, and a(n) = n otherwise.

0, 1, 2, 3, 4, 10, 20, 39, 76, 149, 294, 578, 1136, 2233, 4390, 8631, 16968, 33358, 65580, 128927, 253464, 498297, 979626, 1925894, 3786208, 7443489, 14633514, 28768731, 56557836, 111189778, 218593348, 429743207, 844852900

Offset: 0

Artur Jasinski, Nov 15 2007, Nov 18 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Piezas, Tito III and Weisstein, Eric W., Pentanacci Number.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1).

Cf. A001591, A135055.

a[n_] := a[n] = a[n - 1] + a[n - 2] + a[n - 3] + a[n - 4] + a[n - 5]; a[0] = 0; a[1] = 1; a[2] = 2; a[3] = 3; a[4] = 4; Table[a[n], {n, 0, 50}] (* Artur Jasinski, Nov 18 2007 *)
LinearRecurrence[{1,1,1,1,1},Range[0,4],40] (* Harvey P. Dale, Oct 18 2013 *)

G.f.: x*(x-1)*(2*x^2+2*x+1)/(-1+x^5+x^4+x^3+x^2+x). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009

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

A135056 Pentanacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) if n>=5, and a(n) = n otherwise.

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