cp's OEIS Frontend

A333641 11-gonal (or hendecagonal) square numbers.

%I A333641 #37 Aug 06 2024 22:30:28
%S A333641 0,1,196,29241,1755625,261468900,38941102225,2337990844401,
%T A333641 348201795147556,51858411008887561,3113535139359330841,
%U A333641 463705205422871375236,69060571958250748760481,4146338334574433921200225,617522713934165528806340100,91968930524758079223806760025
%N A333641 11-gonal (or hendecagonal) square numbers.
%C A333641 The 11-gonal square numbers correspond to the nonnegative integer solutions of the Diophantine equation k*(9*k-7)/2 = m^2, equivalent to (18*k-7)^2 - 72*m^2 = 49. Substituting x = 18*k-7 and y = m gives the Pell equation x^2-72*y^2 = 49. The integer solutions (x,y) = (-7,0), (11,1), (119,14), (1451,171), (11243,1325), ... correspond to the following solutions (k,m) = (0,0), (1,1), (7,14), (81,171), (625,1325), ...
%H A333641 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1331714,-1331714,0,-1,1).
%F A333641 a(n) = k*(9*k-7)/2 for n > 1, where k = (A106525(4*n-6) + 7)/18. - _Jinyuan Wang_, Mar 31 2020
%e A333641 1755625 is a term because 625*(9*625-7)/2 = 1325^2 = 1755625; that means that 1755625 is the 625th 11-gonal number and the square of 1325.
%p A333641 for k from 0 to 8000000 do
%p A333641 d:= k*(9*k-7)/2;
%p A333641 if issqr(d) then print(k,sqrt(d),d); else fi; od:
%t A333641 Last /@ Solve[(18*x - 7)^2 - 72*y^2 == 49 && x >= 0 && y >= 0 && y < 10^16, {x, y}, Integers] /. Rule -> (#2^2 &) (* _Amiram Eldar_, Mar 31 2020 *)
%o A333641 (PARI) concat(0, Vec(-x*(1 + 195*x + 29045*x^2 + 394670*x^3 + 29045*x^4 + 195*x^5 + x^6)/(-1 + x + 1331714*x^3 - 1331714*x^4 - x^6 + x^7) + O(x^20))) \\ _Jinyuan Wang_, Mar 31 2020
%Y A333641 Intersection of A000290 (squares) and A051682 (11-gonals).
%Y A333641 Cf. A106525.
%Y A333641 Cf. A001110 (square triangulars), A036353 (square pentagonals), A046177 (square hexagonals), A036354 (square heptagonals), A036428 (square octagonals), A036411 (square 9-gonals), A188896 (only {0,1} are square 10-gonals), this sequence (square 11-gonals), A342709 (square 12-gonals).
%K A333641 nonn,easy
%O A333641 1,3
%A A333641 _Bernard Schott_, Mar 31 2020
%E A333641 More terms from _Amiram Eldar_, Mar 31 2020