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A129445
Numbers k > 0 such that k^2 is a centered triangular number.
10
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
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
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
Prime terms are listed in A129446.
Cf. A125602 (prime CTN), A184481 (semiprime CTN), A125603.
Sequence in context: A074797 A248115 A240285 * A082821 A188893 A227127
KEYWORD
nonn,easy
AUTHOR
Alexander Adamchuk, Apr 15 2007, Apr 26 2007
EXTENSIONS
More terms from Alexander Adamchuk, Apr 26 2007
STATUS
approved