A129445 Numbers k > 0 such that k^2 is a centered triangular number.

1, 2, 8, 19, 79, 188, 782, 1861, 7741, 18422, 76628, 182359, 758539, 1805168, 7508762, 17869321, 74329081, 176888042, 735782048, 1751011099, 7283491399, 17333222948, 72099131942, 171581218381, 713707828021, 1698478960862, 7064979148268, 16813208390239

Offset

1,2

Comments

Corresponding numbers n such that centered triangular number A005448(n) is a perfect square are listed in A129444(n).

Consider Diophantine equation 3*x*(x-1) + 2 - 2*y^2 = 0. Sequence gives solutions for y. - Zak Seidov, Jun 11 2013

Positive values of x (or y) satisfying x^2 - 10xy + y^2 + 15 = 0. - Colin Barker, Feb 09 2014

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

Links

Alexander Adamchuk, Table of n, a(n) for n = 1..100

Tom Beldon and Tony Gardiner, Triangular Numbers and Perfect Squares, The Mathematical Gazette, Vol. 86, No. 507, (2002), pp. 423-431. - Ant King, Dec 07 2010

Index entries for linear recurrences with constant coefficients, signature (0, 10, 0, -1).

Formula

a(n) = sqrt(3*A129444(n)*(A129444(n) - 1)/2 + 1).

G.f.: x*(1-x)*(1+3*x+x^2)/(1-10*x^2+x^4). - Colin Barker, Apr 11 2012

a(n) = 10*a(n-2) - a(n-4), a(1..4) = 1, 2, 8, 19. - Zak Seidov, Jun 11 2013

Mathematica

Do[f = 3n(n-1)/2 + 1; If[IntegerQ[Sqrt[f]], Print[Sqrt[f]]], {n, 150000}]
LinearRecurrence[{0, 10, 0, -1}, {1, 2, 8, 19}, 30] (* T. D. Noe, Jun 13 2013 *)

Crossrefs

Cf. A005448, A129444.

Prime terms are listed in A129446.

Cf. A125602 (prime CTN), A184481 (semiprime CTN), A125603.

Cf. A000290, A249483.

Sequence in context: A074797 A248115 A240285 * A082821 A188893 A227127

Adjacent sequences: A129442 A129443 A129444 * A129446 A129447 A129448

Keyword

nonn,easy

Author

Alexander Adamchuk, Apr 15 2007, Apr 26 2007

Extensions

More terms from Alexander Adamchuk, Apr 26 2007

Status

approved