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A342709 12-gonal (dodecagonal) square numbers. 4
1, 64, 3025, 142129, 6677056, 313679521, 14736260449, 692290561600, 32522920134769, 1527884955772561, 71778070001175616, 3372041405099481409, 158414167969674450625, 7442093853169599697984, 349619996931001511354641, 16424697761903901433970161 (list; graph; refs; listen; history; text; internal format)
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
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
Intersection of A000290 (squares) and A051324 (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).
Sequence in context: A282397 A297100 A221090 * A264094 A181249 A260857
KEYWORD
nonn,easy
AUTHOR
Bernard Schott, Mar 19 2021
STATUS
approved

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Last modified February 23 20:31 EST 2024. Contains 370288 sequences. (Running on oeis4.)