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A129445
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Numbers k > 0 such that k^2 is a centered triangular number.
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9
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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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).
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FORMULA
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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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
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KEYWORD
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nonn,easy
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AUTHOR
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Alexander Adamchuk, Apr 15 2007, Apr 26 2007
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EXTENSIONS
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More terms from Alexander Adamchuk, Apr 26 2007
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STATUS
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approved
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