|
%I #36 Feb 02 2021 22:44:46
%S 1,2,8,19,79,188,782,1861,7741,18422,76628,182359,758539,1805168,
%T 7508762,17869321,74329081,176888042,735782048,1751011099,7283491399,
%U 17333222948,72099131942,171581218381,713707828021,1698478960862,7064979148268,16813208390239
%N Numbers k > 0 such that k^2 is a centered triangular number.
%C Corresponding numbers n such that centered triangular number A005448(n) is a perfect square are listed in A129444(n).
%C Consider Diophantine equation 3*x*(x-1) + 2 - 2*y^2 = 0. Sequence gives solutions for y. - _Zak Seidov_, Jun 11 2013
%C Positive values of x (or y) satisfying x^2 - 10xy + y^2 + 15 = 0. - _Colin Barker_, Feb 09 2014
%C Nonnegative values of x of solutions (x, y) to the Diophantine equation 8*x^2 - 3*y^2 = 5. - _Jon E. Schoenfield_, Feb 02 2021
%H Alexander Adamchuk, <a href="/A129445/b129445.txt">Table of n, a(n) for n = 1..100</a>
%H Tom Beldon and Tony Gardiner, <a href="http://www.jstor.org/stable/3621134">Triangular Numbers and Perfect Squares</a>, The Mathematical Gazette, Vol. 86, No. 507, (2002), pp. 423-431. - _Ant King_, Dec 07 2010
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0, 10, 0, -1).
%F a(n) = sqrt(3*A129444(n)*(A129444(n) - 1)/2 + 1).
%F G.f.: x*(1-x)*(1+3*x+x^2)/(1-10*x^2+x^4). - _Colin Barker_, Apr 11 2012
%F a(n) = 10*a(n-2) - a(n-4), a(1..4) = 1, 2, 8, 19. - _Zak Seidov_, Jun 11 2013
%t Do[f = 3n(n-1)/2 + 1; If[IntegerQ[Sqrt[f]], Print[Sqrt[f]]], {n, 150000}]
%t LinearRecurrence[{0, 10, 0, -1}, {1, 2, 8, 19}, 30] (* _T. D. Noe_, Jun 13 2013 *)
%Y Cf. A005448, A129444.
%Y Prime terms are listed in A129446.
%Y Cf. A125602 (prime CTN), A184481 (semiprime CTN), A125603.
%Y Cf. A000290, A249483.
%K nonn,easy
%O 1,2
%A _Alexander Adamchuk_, Apr 15 2007, Apr 26 2007
%E More terms from _Alexander Adamchuk_, Apr 26 2007
|
