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

0,1

Comments

If p is prime, p divides a(p).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)

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

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)

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

Prog

(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
(PARI) polsym(x^6-x-1, 66) \\ Joerg Arndt, Mar 10 2019

Crossrefs

Column 5 of A306646.

Cf. A001608, A050443.

Cf. A087935.

Sequence in context: A229657 A306550 A339861 * A089804 A270606 A087255

Adjacent sequences: A087933 A087934 A087935 * A087937 A087938 A087939

Keyword

nonn,easy

Author

Benoit Cloitre, Oct 27 2003

Status

approved