This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A154151 #21 Dec 23 2024 14:53:42
%S A154151 0,18,21,111,128,650,749,3791,4368,22098,25461,128799,148400,750698,
%T A154151 864941,4375391,5041248,25501650,29382549,148634511,171254048,
%U A154151 866305418,998141741,5049197999,5817596400,29428882578,33907436661,171524097471,197627023568
%N A154151 Indices k such that 25 plus the k-th triangular number is a perfect square.
%H A154151 Harvey P. Dale, <a href="/A154151/b154151.txt">Table of n, a(n) for n = 1..1000</a>
%H A154151 F. T. Adams-Watters, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2009-October/002504.html">SeqFan Discussion</a>, Oct 2009
%H A154151 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,6,-6,-1,1).
%F A154151 {k: 25+k*(k+1)/2 in A000290}.
%F A154151 Conjectures: (Start)
%F A154151 a(n)= +a(n-1) +6*a(n-2) -6*a(n-3) -a(n-4) +a(n-5).
%F A154151 G.f.: x^2*(-18-3*x+18*x^2+x^3)/( (x-1) * (x^2+2*x-1) * (x^2-2*x-1)).
%F A154151 G.f.: ( 2 + 1/(x-1) + (10+29*x)/(x^2-2*x-1) + (-9+8*x)/(x^2+2*x-1) )/2. (End)
%F A154151 The first conjecture is true for the first 1000 terms of the sequence. - _Harvey P. Dale_, Jun 15 2013
%e A154151 0*(0+1)/2+25 = 5^2. 18*(18+1)/2+25 = 14^2. 21*(21+1)/2+25 = 16^2. 111*(111+1)/2+25 = 79^2.
%t A154151 Join[{0}, Select[Range[0, 10^5], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 25 &]] (* or *) LinearRecurrence[{1,6,-6,-1,1}, {0,18,21,111,128}, 25] (* G. C. Greubel_, Sep 03 2016 *)
%o A154151 (PARI) for(n=1,10^10,if(issquare(25+n*(n+1)/2),print1(n,", ")))
%Y A154151 Cf. A000217, A000290, A006451.
%K A154151 nonn,less,easy
%O A154151 1,2
%A A154151 _R. J. Mathar_, Oct 18 2009