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

Author

Alexander Adamchuk, Apr 15 2007, Apr 26 2007

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A005448, A129444.

Prime terms are listed in A129446.

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

Cf. A000290, A249483.

Programs

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 *)

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

Extensions

More terms from Alexander Adamchuk, Apr 26 2007