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A165517 Indices of the least triangular numbers (A000217) for which three consecutive triangular numbers sum to a perfect square (A000290). 9
0, 5, 14, 63, 152, 637, 1518, 6319, 15040, 62565, 148894, 619343, 1473912, 6130877, 14590238, 60689439, 144428480, 600763525, 1429694574, 5946945823, 14152517272, 58868694717, 140095478158, 582740001359, 1386802264320 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Those perfect squares that can be expressed as the sum of three consecutive triangular numbers correspond to integer solutions of the equation T(k)+T(k+1)+T(k+2)=s^2, or equivalently to 3k^2 + 9k + 8 = 2s^2. Hence solutions occur whenever (3k^2 + 9k + 8)/2 is a perfect square, or equivalently when s>=2 and sqrt(24s^2 - 15) is congruent to 3 mod 6. This sequence returns the index of the smallest of the 3 triangular numbers, the values of s^2 are given in A165516 and, with the exception of the first term, the values of s are in A129445.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

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

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

FORMULA

a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) - a(n-4) + a(n-5).

G.f.: x(x^3 + x^2 - 9x - 5)/((x-1)(x^4 - 10x^2 + 1)).

a(n) = 10*a(n-2) - a(n-4) + 12. - Zak Seidov, Sep 25 2009

EXAMPLE

The fourth perfect square that can be expressed as the sum of three consecutive triangular numbers is 6241 = T(63) + T(64) + T(65). Hence a(4)=63.

MATHEMATICA

TriangularNumber[ n_ ]:=1/2 n (n+1); Select[ Range[ 0, 10^7 ], IntegerQ[ Sqrt[ TriangularNumber[ # ]+TriangularNumber[ #+1 ]+TriangularNumber[ #+2 ] ] ] & ]

CoefficientList[Series[x*(x^3 + x^2 - 9*x - 5)/((x - 1)*(x^4 - 10*x^2 + 1)), {x, 0, 50}], x] (* or *) LinearRecurrence[{1, 10, -10, -1, 1}, {0, 5, 14, 63, 152}, 50] (* G. C. Greubel, Feb 17 2017 *)

PROG

(PARI) x='x+O('x^50); concat([0], Vec(x*(x^3 + x^2 - 9*x - 5)/((x - 1)*(x^4 - 10*x^2 + 1)))) \\ G. C. Greubel, Feb 17 2017

(MAGMA) I:=[0, 5, 14, 63, 152]; [n le 5 select I[n] else Self(n-1) + 10*Self(n-2) - 10*Self(n-3) - Self(n-4) + Self(n-5): n in [1..50]]; // G. C. Greubel, Oct 21 2018

CROSSREFS

Cf. A000290, A129445, A000217, A165516.

Sequence in context: A279511 A281698 A268814 * A197788 A197661 A004030

Adjacent sequences:  A165514 A165515 A165516 * A165518 A165519 A165520

KEYWORD

easy,nonn

AUTHOR

Ant King, Sep 25 2009, Oct 01 2009

EXTENSIONS

a(1) = 0 added by N. J. A. Sloane, Sep 28 2009, at the suggestion of Alexander R. Povolotsky

More terms from Zak Seidov, Sep 25 2009

STATUS

approved

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Last modified March 31 19:37 EDT 2020. Contains 333151 sequences. (Running on oeis4.)