OFFSET
1,2
COMMENTS
The dodecagonal square numbers k correspond to the nonnegative integer solutions of the Diophantine equation k = d*(5*d-4) = c^2, equivalent to (5*d-2)^2 - 5*c^2 = 4. Substituting x = 5*d-2 and y = c gives the Pell-Fermat's equation x^2 - 5*y^2 = 4.
The solutions x are in A342710, while corresponding solutions y that are also the indices c of the squares which are 12-gonal are in A033890.
The indices d of the corresponding 12-gonal which are squares are in A081068.
LINKS
Index entries for linear recurrences with constant coefficients, signature (48,-48,1).
FORMULA
G.f.: x*(1 + 16*x + x^2)/((1 - x)*(1 - 47*x + x^2)). - Stefano Spezia, Mar 20 2021
a(n) = 48*a(n-1) - 48*a(n-2) + a(n-3). - Kevin Ryde, Mar 20 2021
a(n) = 9*A161582(n) + 1. - Hugo Pfoertner, Mar 19 2021
a(n) = A033890(n-1)^2.
EXAMPLE
142129 = 169*(5*169-4) = 377^2, so 142129 is the 169th 12-gonal number and the 377th square, hence 142129 is a term.
MAPLE
With(combinat);
E := seq(fibonacci(4*n-2)^2, n=1..16);
MATHEMATICA
Table[Fibonacci[4*n - 2]^2, {n, 1, 16}] (* Amiram Eldar, Mar 19 2021 *)
PROG
(PARI) a(n) = fibonacci(4*n-2)^2; \\ Michel Marcus, Mar 21 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bernard Schott, Mar 19 2021
STATUS
approved