A342709 12-gonal (dodecagonal) square numbers.

1, 64, 3025, 142129, 6677056, 313679521, 14736260449, 692290561600, 32522920134769, 1527884955772561, 71778070001175616, 3372041405099481409, 158414167969674450625, 7442093853169599697984, 349619996931001511354641, 16424697761903901433970161

Offset: 1

Author

Bernard Schott, Mar 19 2021

Keywords

• nonn
• easy

Comments

The 12-gonal 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.

Examples

			142129 = 169*(5*169-4) = 377^2, so 142129 is the 169th 12-gonal number and the 377th square, hence 142129 is a term.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (48,-48,1).

Crossrefs

Intersection of A000290 (squares) and A051624 (12-gonal numbers).

Similar for n-gonal squares: A001110 (triangular), A036353 (pentagonal), A046177 (hexagonal), A036354 (heptagonal), A036428 (octagonal), A036411 (9-gonal), A188896 (there are no 10-gonal squares > 1), A333641 (11-gonal), this sequence (12-gonal).

Programs

Maple

with(combinat):
seq(fibonacci(4*n-2)^2, n=1..16);

Mathematica

Table[Fibonacci[4*n - 2]^2, {n, 1, 16}] (* Amiram Eldar, Mar 19 2021 *)

PARI

a(n) = fibonacci(4*n-2)^2; \\ Michel Marcus, Mar 21 2021

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.