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A154152 Indices k such that 26 plus the k-th triangular number is a perfect square. 2

%I

%S 4,10,37,67,220,394,1285,2299,7492,13402,43669,78115,254524,455290,

%T 1483477,2653627,8646340,15466474,50394565,90145219,293721052,

%U 525404842,1711931749,3062283835,9977869444,17848298170,58155284917,104027505187,338953840060

%N Indices k such that 26 plus the k-th triangular number is a perfect square.

%H Colin Barker, <a href="/A154152/b154152.txt">Table of n, a(n) for n = 1..500</a>

%H F. T. Adams-Watters, <a href="http://list.seqfan.eu/pipermail/seqfan/2009-October/002504.html">SeqFan Discussion</a>, Oct 2009

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,6,-6,-1,1).

%F {k: 26+k*(k+1)/2 in A000290}.

%F a(n) = +a(n-1) +6*a(n-2) -6*a(n-3) -a(n-4) +a(n-5).

%F G.f.: x*(-4-6*x-3*x^2+6*x^3+5*x^4)/((x-1) * (x^2-2*x-1) * (x^2+2*x-1)).

%F G.f.: ( 10 + (-3-6*x)/(x^2+2*x-1) + 1/(x-1) + (12+27*x)/(x^2-2*x-1) )/2.

%e 4*(4+1)/2+26 = 6^2. 10*(10+1)/2+26 = 9^2. 37*(37+1)/2+26 = 27^2. 67*(67+1)/2+26 = 48^2.

%t Join[{4, 10}, Select[Range[0,10^5], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 26 &]] (* or *) LinearRecurrence[{1,6,-6,-1,1}, {4,10,37,67,220}, 25] (* _G. C. Greubel_, Sep 03 2016 *)

%o (PARI) Vec(x*(-4-6*x-3*x^2+6*x^3+5*x^4)/((x-1)*(x^2-2*x-1)*(x^2+2*x-1)) + O(x^40)) \\ _Colin Barker_, Jul 11 2015

%Y Cf. A000217, A000290, A006451.

%K nonn,less,easy

%O 1,1

%A _R. J. Mathar_, Oct 18 2009

%E Extended by _D. S. McNeil_, Dec 05 2010

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Last modified March 26 12:43 EDT 2019. Contains 321497 sequences. (Running on oeis4.)