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A154154 Numbers k such that 30 plus the k-th triangular number is a perfect square.

%I A154154 #18 Dec 23 2024 14:53:42
%S A154154 3,13,34,84,203,493,1186,2876,6915,16765,40306,97716,234923,569533,
%T A154154 1369234,3319484,7980483,19347373,46513666
%N A154154 Numbers k such that 30 plus the k-th triangular number is a perfect square.
%H A154154 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
%F A154154 {k: 30+k*(k+1)/2 in A000290}.
%F A154154 Conjectures: (Start)
%F A154154 a(n) = +a(n-1) +6*a(n-2) -6*a(n-3) -a(n-4) +a(n-5).
%F A154154 G.f.: x*(-3-10*x-3*x^2+10*x^3+4*x^4)/((x-1) * (x^2-2*x-1) * (x^2+2*x-1)).
%F A154154 G.f.: ( 8 + (-5-2*x)/(x^2+2*x-1) + (12+29*x)/(x^2-2*x-1) + 1/(x-1) )/2. (End)
%e A154154 3, 13, 34, and 84 are terms:
%e A154154    3* (3+1)/2 + 30 =  6^2,
%e A154154   13*(13+1)/2 + 30 = 11^2,
%e A154154   34*(34+1)/2 + 30 = 25^2,
%e A154154   84*(84+1)/2 + 30 = 60^2.
%t A154154 Position[Accumulate[Range[8*10^6]],_?(IntegerQ[Sqrt[#+30]]&)]//Flatten (* _Harvey P. Dale_, May 30 2016 *)
%t A154154 Join[{3, 13}, Select[Range[0, 10^5], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 30 &]] (* _G. C. Greubel_, Sep 03 2016 *)
%o A154154 (PARI) {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 30), print1(n, ", ") ) );}
%Y A154154 Cf. A000217, A000290, A006451.
%K A154154 nonn,less
%O A154154 1,1
%A A154154 _R. J. Mathar_, Oct 18 2009