%I #25 Jan 05 2023 19:08:40

%S 0,1,15,128,945,6655,46080,317057,2176335,14925184,102320625,

%T 701373311,4807434240,32951037313,225850798095,1548007091840,

%U 10610205501105,72723448842367,498453982018560,3416454544730369,23416728143799375

%N a(n) = F(4n) - 2F(2n) where F(n) = Fibonacci numbers A000045.

%C a(n) is a divisibility sequence; that is, if h|k then a(h)|a(k).

%H Michael De Vlieger, <a href="/A127595/b127595.txt">Table of n, a(n) for n = 0..1196</a>

%H E. L. Roettger and H. C. Williams, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Roettger/roettger12.html">Appearance of Primes in Fourth-Order Odd Divisibility Sequences</a>, J. Int. Seq., Vol. 24 (2021), Article 21.7.5.

%H Hugh Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory vol. 7 (5) (2011) 1255-1277.

%H H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/p33/p33.Abstract.html">Odd and even linear divisibility sequences of order 4</a>, INTEGERS, 2015, #A33.

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (10,-23,10,-1).

%F a(n) = F(2n)*(L(2n)-2) = A001906(n)*A004146(n), where L(n) are the Lucas numbers A000032.

%F a(2n) = 5*(F(2n))^3*L(2n), a(2n+1) = F(2n+1)*L(2n+1)^3.

%F a(n) = [(Phi^(2n))-1]^2*[(Phi^(4n))-1]/[sqrt(5)*(Phi^(4n))].

%F G.f.: A(x)=x*(1+(r+2)*x+x^2)/((1-r*x+x^2)*(1-(r^2-2)*x+x^2)) at r=3. The case r=2 is A000578.

%F a(n) = -a(-n) for all n in Z. - _Michael Somos_, Dec 30 2022

%e G.f. = x + 15*x^2 + 128*x^3 + 945*x^4 + 6655*x^5 + ... - _Michael Somos_, Dec 30 2022

%t With[{r = 3}, CoefficientList[Series[x (1 + (r + 2) x + x^2)/((1 - r x + x^2)*(1 - (r^2 - 2)*x + x^2)), {x, 0, 20}], x]] (* _Michael De Vlieger_, Nov 09 2021 *)

%o (PARI) {a(n) = my(w = quadgen(5)^(2*n)); imag(w^2 - 2*w)}; /* _Michael Somos_, Dec 30 2022 */

%Y Cf. A000032, A000045, A001906, A004146.

%K easy,nonn

%O 0,3

%A _Peter Bala_, Apr 10 2007