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A087935 Perrin sequence of order 5. 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 (list; graph; refs; listen; history; text; internal format)
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: A229656 A216722 A036297 * A243829 A318952 A089877

Adjacent sequences:  A087932 A087933 A087934 * A087936 A087937 A087938

KEYWORD

nonn,easy

AUTHOR

Benoit Cloitre, Oct 27 2003

STATUS

approved

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Last modified October 22 00:52 EDT 2019. Contains 328315 sequences. (Running on oeis4.)