cp's OEIS Frontend

A372015 Product of Fibonacci and self-convolution of Fibonacci numbers: a(n) = A000045(n+1)*A001629(n+1).

%I A372015 #22 Jul 30 2024 02:34:31
%S A372015 0,1,4,15,50,160,494,1491,4420,12925,37380,107136,304764,861445,
%T A372015 2421700,6775755,18879734,52413856,145038890,400183575,1101277060,
%U A372015 3023462521,8282790024,22646131200,61805595000,168399404425,458128878724,1244567262471,3376576740410,9149594423200
%N A372015 Product of Fibonacci and self-convolution of Fibonacci numbers: a(n) = A000045(n+1)*A001629(n+1).
%C A372015 Conjecture: a(n) is the total number of pairs of adjacent parts that are the same color in all n-color compositions of n+1. - _John Tyler Rascoe_, Jul 30 2024
%H A372015 John Tyler Rascoe, <a href="/A372015/b372015.txt">Table of n, a(n) for n = 0..500</a>
%H A372015 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,-5,5,-1).
%F A372015 a(n) = F(n+1)*((n+2)*F(n) + (n)*F(n+2))/5 where F(n) = A000045(n) is the Fibonacci numbers.
%F A372015 G.f.: x*(1-x)/((1+x)*(1-3*x+x^2)^2).
%p A372015 a := proc(n) option remember; if n < 3 then return n^2 fi;
%p A372015 -((2 - 2*n^2 + n)*a(n - 1) + (1 - 2*n^2 + 3*n)*a(n - 2) + n^2*a(n - 3))/(n - 1)^2 end: seq(a(n), n = 0..29);  # _Peter Luschny_, Apr 16 2024
%t A372015 CoefficientList[Series[x(1-x)/((1+x)*(1-3*x+x^2)^2),{x,0,29}],x] (* _Stefano Spezia_, Apr 16 2024 *)
%o A372015 (PARI)
%o A372015 A_x(N)= {my(x='x+O('x^N)); concat([0],Vec(x*(1-x)/((1+x)*(1-3*x+x^2)^2)))}
%o A372015 A_x(40) \\ _John Tyler Rascoe_, Jul 29 2024
%Y A372015 Cf. A000045, A001629, A088305.
%K A372015 nonn,easy
%O A372015 0,3
%A A372015 _Vladimir Kruchinin_, Apr 15 2024