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A087936 Perrin sequence of order 6.

%I A087936 #47 Apr 24 2025 10:46:33
%S A087936 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,
%T A087936 29,11,31,80,110,85,40,42,111,190,195,125,82,153,301,385,320,207,235,
%U A087936 454,686,705,527,442,689,1140,1391,1232,969,1131,1829,2531,2623,2201,2100
%N A087936 Perrin sequence of order 6.
%C A087936 If p is prime, p divides a(p).
%H A087936 Seiichi Manyama, <a href="/A087936/b087936.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H A087936 Sadjia Abbad and Hacène Belbachir, <a href="https://math.colgate.edu/~integers/z38/z38.pdf">The r-Fibonacci polynomial and its companion sequences linked with some classical sequences</a>, Integers (2025), Vol. 25, Art. No. A38. See p. 17.
%H A087936 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,1,1).
%F A087936 a(n) = a(n-5) + a(n-6) with a(0)=6, a(1)=a(2)=a(3)=a(4)=0, a(5)=5.
%F A087936 a(n) = Sum_{i=1..6} (x_i)^n where x_i are the roots of x^6 = x+1.
%F A087936 G.f.: (x^5-6) / (x^6+x^5-1). - _Colin Barker_, Jun 16 2013
%F A087936 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
%F A087936 From _Aleksander Bosek_, Mar 06 2019: (Start)
%F A087936 a((s+6)*n+m) = Sum_{l=0..n} binomial(n-l,l)*a(s*n+l+m) for all s > 0, m > 0.
%F A087936 a(m) = Sum_{l=0..n}(-1)^{n-l} binomial(n-l,l)*a(m+n+5*l)for all m > 0. (End)
%p A087936 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
%o A087936 (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
%o A087936 (PARI) polsym(x^6-x-1,66) \\ _Joerg Arndt_, Mar 10 2019
%Y A087936 Column 5 of A306646.
%Y A087936 Cf. A001608, A050443.
%Y A087936 Cf. A087935.
%K A087936 nonn,easy
%O A087936 0,1
%A A087936 _Benoit Cloitre_, Oct 27 2003