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A342709 12-gonal (dodecagonal) square numbers.

%I A342709 #56 Dec 13 2024 18:21:48
%S A342709 1,64,3025,142129,6677056,313679521,14736260449,692290561600,
%T A342709 32522920134769,1527884955772561,71778070001175616,
%U A342709 3372041405099481409,158414167969674450625,7442093853169599697984,349619996931001511354641,16424697761903901433970161
%N A342709 12-gonal (dodecagonal) square numbers.
%C A342709 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.
%C A342709 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.
%C A342709 The indices d of the corresponding 12-gonal which are squares are in A081068.
%H A342709 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (48,-48,1).
%F A342709 G.f.: x*(1 + 16*x + x^2)/((1 - x)*(1 - 47*x + x^2)). - _Stefano Spezia_, Mar 20 2021
%F A342709 a(n) = 48*a(n-1) - 48*a(n-2) + a(n-3). - _Kevin Ryde_, Mar 20 2021
%F A342709 a(n) = 9*A161582(n) + 1. - _Hugo Pfoertner_, Mar 19 2021
%F A342709 a(n) = A033890(n-1)^2.
%e A342709 142129 = 169*(5*169-4) = 377^2, so 142129 is the 169th 12-gonal number and the 377th square, hence 142129 is a term.
%p A342709 with(combinat):
%p A342709 seq(fibonacci(4*n-2)^2, n=1..16);
%t A342709 Table[Fibonacci[4*n - 2]^2, {n, 1, 16}] (* _Amiram Eldar_, Mar 19 2021 *)
%o A342709 (PARI) a(n) = fibonacci(4*n-2)^2; \\ _Michel Marcus_, Mar 21 2021
%Y A342709 Intersection of A000290 (squares) and A051624 (12-gonal numbers).
%Y A342709 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).
%K A342709 nonn,easy
%O A342709 1,2
%A A342709 _Bernard Schott_, Mar 19 2021