A087936 Perrin sequence of order 6.
6, 0, 0, 0, 0, 5, 6, 0, 0, 0, 5, 11, 6, 0, 0, 5, 16, 17, 6, 0, 5, 21, 33, 23, 6, 5, 26, 54, 56, 29, 11, 31, 80, 110, 85, 40, 42, 111, 190, 195, 125, 82, 153, 301, 385, 320, 207, 235, 454, 686, 705, 527, 442, 689, 1140, 1391, 1232, 969, 1131, 1829, 2531, 2623, 2201, 2100
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
- Sadjia Abbad and Hacène Belbachir, The r-Fibonacci polynomial and its companion sequences linked with some classical sequences, Integers (2025), Vol. 25, Art. No. A38. See p. 17.
- Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1,1).
Programs
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GAP
Concatenation([6],List([1..65],n->n*Sum([1..Int(n/5)],k->Binomial(k,n-5*k)/k))); # Muniru A Asiru, Mar 09 2019
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Maple
a:=n->n*add(binomial(k,n-5*k)/k,k=1..floor(n/5)): 6,seq(a(n),n=1..65); # Muniru A Asiru, Mar 09 2019
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PARI
polsym(x^6-x-1,66) \\ Joerg Arndt, Mar 10 2019
Formula
a(n) = a(n-5) + a(n-6) with a(0)=6, a(1)=a(2)=a(3)=a(4)=0, a(5)=5.
a(n) = Sum_{i=1..6} (x_i)^n where x_i are the roots of x^6 = x+1.
G.f.: (x^5-6) / (x^6+x^5-1). - Colin Barker, Jun 16 2013
a(0) = 6 and a(n) = n*Sum_{k=1..floor(n/5)} binomial(k,n-5*k)/k for n > 0. - Seiichi Manyama, Mar 04 2019
From Aleksander Bosek, Mar 06 2019: (Start)
a((s+6)*n+m) = Sum_{l=0..n} binomial(n-l,l)*a(s*n+l+m) for all s > 0, m > 0.
a(m) = Sum_{l=0..n}(-1)^{n-l} binomial(n-l,l)*a(m+n+5*l)for all m > 0. (End)
Comments