OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (4,10,4,-1).
FORMULA
G.f.: 1/((1+x)^2*(1-6*x+x^2)).
Self-convolution of A026933.
Self-convolution 4th power of A204061.
a(n) = Pell(n-1)^2 + a(n-2) where Pell(n) = A000129(n).
a(n) = (1/8)*(A001109(n+2) + (-1)^n*(n+2)). - Bruno Berselli, Jan 10 2012
a(n) = (1/16)*(A000129(2*n+4) + 2*(-1)^n*(n+2)). - G. C. Greubel, May 25 2021
EXAMPLE
G.f.: A(x) = 1 + 4*x + 26*x^2 + 148*x^3 + 867*x^4 + 5048*x^5 + ...
where
log(A(x)) = 2^2*x + 6^2*x^2/2 + 14^2*x^3/3 + 34^2*x^4/4 + 82^2*x^5/5 + 198^2*x^6/6 + 478^2*x^7/7 + ... + A002203(n)^2*x^n/n + ...
MATHEMATICA
LinearRecurrence[{4, 10, 4, -1}, {1, 4, 26, 148}, 30] (* Vincenzo Librandi, Feb 12 2012 *)
Table[(Fibonacci[2*n+4, 2] + 2*(-1)^n*(n+2))/16, {n, 0, 30}] (* G. C. Greubel, May 25 2021 *)
PROG
(PARI) {A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)), n)}
{a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^2*x^k/k)+x*O(x^n)), n)}
(Magma) I:=[1, 4, 26, 148]; [n le 4 select I[n] else 4*Self(n-1) +10*Self(n-2) +4*Self(n-3) -Self(n-4): n in [1..31]]; // G. C. Greubel, May 25 2021
(Sage) [(lucas_number1(2*n+4, 2, -1) +2*(-1)^n*(n+2))/16 for n in (0..30)] # G. C. Greubel, May 25 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Jan 10 2012
STATUS
approved