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%I #20 Sep 08 2022 08:45:47
%S 0,5,14,63,152,637,1518,6319,15040,62565,148894,619343,1473912,
%T 6130877,14590238,60689439,144428480,600763525,1429694574,5946945823,
%U 14152517272,58868694717,140095478158,582740001359,1386802264320
%N Indices of the least triangular numbers (A000217) for which three consecutive triangular numbers sum to a perfect square (A000290).
%C 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.
%H G. C. Greubel, <a href="/A165517/b165517.txt">Table of n, a(n) for n = 1..1000</a>
%H Tom Beldon and Tony Gardiner, <a href="http://www.jstor.org/stable/3621134">Triangular Numbers and Perfect Squares</a>, The Mathematical Gazette, Vol. 86, No. 507, (2002), pp. 423-431.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,10,-10,-1,1).
%F a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) - a(n-4) + a(n-5).
%F G.f.: x(x^3 + x^2 - 9x - 5)/((x-1)(x^4 - 10x^2 + 1)).
%F a(n) = 10*a(n-2) - a(n-4) + 12. - _Zak Seidov_, Sep 25 2009
%e 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.
%t TriangularNumber[ n_ ]:=1/2 n (n+1);Select[ Range[ 0,10^7 ], IntegerQ[ Sqrt[ TriangularNumber[ # ]+TriangularNumber[ #+1 ]+TriangularNumber[ #+2 ] ] ] & ]
%t 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 *)
%o (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
%o (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
%Y Cf. A000290, A129445, A000217, A165516.
%K easy,nonn
%O 1,2
%A _Ant King_, Sep 25 2009, Oct 01 2009
%E a(1) = 0 added by _N. J. A. Sloane_, Sep 28 2009, at the suggestion of _Alexander R. Povolotsky_
%E More terms from _Zak Seidov_, Sep 25 2009
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