A087935 Perrin sequence of order 5.

5, 0, 0, 0, 4, 5, 0, 0, 4, 9, 5, 0, 4, 13, 14, 5, 4, 17, 27, 19, 9, 21, 44, 46, 28, 30, 65, 90, 74, 58, 95, 155, 164, 132, 153, 250, 319, 296, 285, 403, 569, 615, 581, 688, 972, 1184, 1196, 1269, 1660, 2156, 2380, 2465, 2929, 3816, 4536, 4845, 5394, 6745, 8352, 9381

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 (0,0,0,1,1).

Formula

a(n) = a(n-4) + a(n-5), with a(0)=5, a(1)=a(2)=a(3)=0.

a(n) = (x_1)^n + (x_2)^n + (x_3)^n + (x_4)^n + (x_5)^n where (x_i) 1 <= i <= 5 are the roots of x^5=x+1.

G.f.: (5 - x^4)/(1 -x^4 -x^5). - Colin Barker, Jun 16 2013

a(0) = 5 and a(n) = n*Sum_{k=1..floor(n/4)} binomial(k,n-4*k)/k for n > 0. - Seiichi Manyama, Mar 04 2019

From Aleksander Bosek, Mar 06 2019: (Start)

a((s+5)*n + m) = Sum_{j=0..n} binomial(n-j,j)*a(s*n+j+m) for all s > 0, m > 0.

a(m) = Sum_{j=0..n} (-1)^(n-j)*binomial(n-j,j)*a(m+n+4*j) for all m > 0. (End)

Maple

seq(coeff(series((x^4-5)/(x^5+x^4-1), x, n+1), x, n), n = 0 .. 60); # Muniru A Asiru, Mar 06 2019

Mathematica

LinearRecurrence[{0, 0, 0, 1, 1}, {5, 0, 0, 0, 4}, 60] (* Harvey P. Dale, Oct 03 2016 *)

Prog

(GAP) a:=[5, 0, 0, 0, 4];; for n in [6..60] do a[n]:=a[n-4]+a[n-5]; od; Print(a); # Muniru A Asiru, Mar 06 2019
(PARI) my(x='x+O('x^60)); Vec((5-x^4)/(1-x^4-x^5)) \\ G. C. Greubel, Mar 06 2019
(PARI) polsym(x^5-x-1, 66) \\ Joerg Arndt, Mar 10 2019
(Magma) I:=[5, 0, 0, 0, 4]; [n le 5 select I[n] else Self(n-4) +Self(n-5): n in [1..60]]; // G. C. Greubel, Mar 06 2019
(Sage) ((5-x^4)/(1-x^4-x^5)).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Mar 06 2019

Crossrefs

Column 4 of A306646.

Cf. A001608, A050443.

Cf. A087936.

Sequence in context: A368865 A036297 A336530 * A243829 A329639 A318952

Adjacent sequences: A087932 A087933 A087934 * A087936 A087937 A087938

Keyword

nonn,easy

Author

Benoit Cloitre, Oct 27 2003

Status

approved