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

3, 13, 34, 84, 203, 493, 1186, 2876, 6915, 16765, 40306, 97716, 234923, 569533, 1369234, 3319484, 7980483, 19347373, 46513666

Offset: 1

Author

R. J. Mathar, Oct 18 2009

Keywords

• nonn
• less

Examples

			3, 13, 34, and 84 are terms:
   3* (3+1)/2 + 30 =  6^2,
  13*(13+1)/2 + 30 = 11^2,
  34*(34+1)/2 + 30 = 25^2,
  84*(84+1)/2 + 30 = 60^2.
		

Links

• F. T. Adams-Watters, SeqFan Discussion, Oct 2009

Crossrefs

Cf. A000217, A000290, A006451.

Programs

Mathematica

Position[Accumulate[Range[8*10^6]],?(IntegerQ[Sqrt[#+30]]&)]//Flatten (* _Harvey P. Dale, May 30 2016 *)
Join[{3, 13}, Select[Range[0, 10^5], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 30 &]] (* G. C. Greubel, Sep 03 2016 *)

PARI

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

Formula

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

Conjectures: (Start)

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

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)).

G.f.: ( 8 + (-5-2*x)/(x^2+2*x-1) + (12+29*x)/(x^2-2*x-1) + 1/(x-1) )/2. (End)