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%I #55 Sep 08 2022 08:45:11
%S 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,
%T 74,58,95,155,164,132,153,250,319,296,285,403,569,615,581,688,972,
%U 1184,1196,1269,1660,2156,2380,2465,2929,3816,4536,4845,5394,6745,8352,9381
%N Perrin sequence of order 5.
%C If p is prime, p divides a(p).
%H Seiichi Manyama, <a href="/A087935/b087935.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1,1).
%F a(n) = a(n-4) + a(n-5), with a(0)=5, a(1)=a(2)=a(3)=0.
%F 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.
%F G.f.: (5 - x^4)/(1 -x^4 -x^5). - _Colin Barker_, Jun 16 2013
%F 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
%F From _Aleksander Bosek_, Mar 06 2019: (Start)
%F a((s+5)*n + m) = Sum_{j=0..n} binomial(n-j,j)*a(s*n+j+m) for all s > 0, m > 0.
%F a(m) = Sum_{j=0..n} (-1)^(n-j)*binomial(n-j,j)*a(m+n+4*j) for all m > 0. (End)
%p 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
%t LinearRecurrence[{0,0,0,1,1},{5,0,0,0,4},60] (* _Harvey P. Dale_, Oct 03 2016 *)
%o (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
%o (PARI) my(x='x+O('x^60)); Vec((5-x^4)/(1-x^4-x^5)) \\ _G. C. Greubel_, Mar 06 2019
%o (PARI) polsym(x^5-x-1,66) \\ _Joerg Arndt_, Mar 10 2019
%o (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
%o (Sage) ((5-x^4)/(1-x^4-x^5)).series(x, 60).coefficients(x, sparse=False) # _G. C. Greubel_, Mar 06 2019
%Y Column 4 of A306646.
%Y Cf. A001608, A050443.
%Y Cf. A087936.
%K nonn,easy
%O 0,1
%A _Benoit Cloitre_, Oct 27 2003
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