A154154 - OEIS

%I #17 Jan 02 2023 12:30:47

%S 3,13,34,84,203,493,1186,2876,6915,16765,40306,97716,234923,569533,

%T 1369234,3319484,7980483,19347373,46513666

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

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

%F {k: 30+k*(k+1)/2 in A000290}.

%F Conjectures: (Start)

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

%F 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 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 3, 13, 34, and 84 are terms:

%e    3* (3+1)/2 + 30 =  6^2,

%e   13*(13+1)/2 + 30 = 11^2,

%e   34*(34+1)/2 + 30 = 25^2,

%e   84*(84+1)/2 + 30 = 60^2.

%t Position[Accumulate[Range[8*10^6]],_?(IntegerQ[Sqrt[#+30]]&)]//Flatten (* _Harvey P. Dale_, May 30 2016 *)

%t Join[{3, 13}, Select[Range[0, 10^5], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 30 &]] (* _G. C. Greubel_, Sep 03 2016 *)

%o (PARI) {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 30), print1(n, ", ") ) );}

%Y Cf. A000217, A000290, A006451.

%K nonn,less

%O 1,1

%A _R. J. Mathar_, Oct 18 2009