A395431 - OEIS

%I #50 May 06 2026 09:27:52

%S 0,0,1,8,50,280,1475,7472,36836,178000,847045,3982200,18538134,

%T 85599912,392560775,1789800160,8119213000,36670415648,164983381129,

%U 739737725800,3306651932410,14740349833400,65546786425931,290816440554768,1287647063875500,5690631701996400

%N Self-convolution of A001076.

%H Alois P. Heinz, <a href="/A395431/b395431.txt">Table of n, a(n) for n = 0..1591</a>

%H D. Dmytryshyn, D. Gray, V. Khamitov, and Alex Stokolos, <a href="https://arxiv.org/abs/2603.08636">Convolved numbers of k-section of the Fibonacci sequence: properties, consequences</a>, arXiv:2603.08636 [math.CA], 2026; see also <a href="https://doi.org/10.15276/hait.09.2026.09">alternate link</a>, Herald of Advanced Information Technology. 2026. Vol. 9, No. 2. P. 129-139.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-14,-8,-1).

%F a(n) = 8*a(n-1) - 14*a(n-2) - 8*a(n-3) - a(n-4) for n>=4, a(0)=a(1)=0, a(2)=1, a(3)=8.

%F a(n) = (4*(n-1)*a(n-1)+n*a(n-2))/(n-2) for n>=3, a(0)=a(1)=0, a(2)=1.

%F G.f.: z^2/(1-4*z-z^2)^2.

%F a(n) = ((n-1)*Fibonacci(3*n+3) + (n+1)*Fibonacci(3*n-3))/40.

%F Binet formula: a(n) = (1/(q^{3*(n+2)}*(q^3+q^{-3})^3))*(n*((-1)^{3*n+1}+q^{6*(n+2)})+(n+2)*((-1)^{3*n+1}q^6+q^{6*(n+1)})), where q=(1+sqrt(5))/2 is the golden ratio A001622.

%F Gegenbauer polynomials: a(n) = (-i)^{n-1} * C_{n-1}^{(2)}(2i).

%F E.g.f.: exp(2*x)*(10*x*cosh(sqrt(5)*x) + sqrt(5)*(5*x - 2)*sinh(sqrt(5)*x))/50. - _Stefano Spezia_, Apr 26 2026

%p a:= proc(n) option remember; `if`(n<4, [0$2,1,8][n+1],

%p       8*a(n-1)-14*a(n-2)-8*a(n-3)-a(n-4))

%p     end:

%p seq(a(n), n=0..25);

%p # Alternative:

%p a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|-8|-14|8>>^n)[2,4]:

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Apr 27 2026

%t a[n_]:=((n-1)*Fibonacci[3*n+3]+(n+1)*Fibonacci[3*(n-1)])/40; Array[a,26,0] (* _Stefano Spezia_, Apr 26 2026 *)

%Y Cf. A000045 (Fibonacci), A001076 (F(3*n)/2), A001622, A001629.

%K nonn,easy

%O 0,4

%A _Alex Stokolos_, Apr 22 2026