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A087936 Perrin sequence of order 6. 4

%I #43 Mar 18 2019 19:50:03

%S 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 29,11,31,80,110,85,40,42,111,190,195,125,82,153,301,385,320,207,235,

%U 454,686,705,527,442,689,1140,1391,1232,969,1131,1829,2531,2623,2201,2100

%N Perrin sequence of order 6.

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

%H 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 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,1,1).

%F 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 a(n) = Sum_{i=1..6} (x_i)^n where x_i are the roots of x^6 = x+1.

%F G.f.: (x^5-6) / (x^6+x^5-1). - _Colin Barker_, Jun 16 2013

%F 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 From _Aleksander Bosek_, Mar 06 2019: (Start)

%F 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 a(m) = Sum_{l=0..n}(-1)^{n-l} binomial(n-l,l)*a(m+n+5*l)for all m > 0. (End)

%p 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 (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 (PARI) polsym(x^6-x-1,66) \\ _Joerg Arndt_, Mar 10 2019

%Y Column 5 of A306646.

%Y Cf. A001608, A050443.

%Y Cf. A087935.

%K nonn,easy

%O 0,1

%A _Benoit Cloitre_, Oct 27 2003

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)