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A129445 Numbers k > 0 such that k^2 is a centered triangular number. 9
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 (list; graph; refs; listen; history; text; internal format)
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

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

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Last modified June 4 08:18 EDT 2020. Contains 334825 sequences. (Running on oeis4.)