A154148 Numbers k such that 21 plus the k-th triangular number is a perfect square.
5, 7, 40, 50, 237, 295, 1384, 1722, 8069, 10039, 47032, 58514, 274125, 341047, 1597720, 1987770, 9312197, 11585575, 54275464, 67525682
Offset: 1
Examples
5, 7, 40, and 50 are terms: 5* (5+1)/2 + 21 = 6^2, 7* (7+1)/2 + 21 = 7^2, 40*(40+1)/2 + 21 = 29^2, 50*(50+1)/2 + 21 = 36^2.
Links
- F. T. Adams-Watters, SeqFan Discussion, Oct 2009
Programs
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Mathematica
Select[Range[68*10^6],IntegerQ[Sqrt[21+(#(#+1))/2]]&] (* Harvey P. Dale, Mar 07 2017 *)
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PARI
{for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 21), print1(n, ", ") ) );}
Formula
{k: 21+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*(-5-2*x-3*x^2+2*x^3+6*x^4)/((x-1) * (x^2-2*x-1) * (x^2+2*x-1)).
G.f.: ( 12 + (12+25*x)/(x^2-2*x-1) + 1/(x-1) + (-1-10*x)/(x^2+2*x-1) )/2. (End)