A154150 Numbers k such that 24 plus the k-th triangular number is a perfect square.
1, 15, 24, 94, 145, 551, 848, 3214, 4945, 18735, 28824, 109198, 168001, 636455, 979184, 3709534, 5707105, 21620751, 33263448, 126014974
Offset: 1
Examples
1, 5, 24, and 94 are terms: 1* (1+1)/2 + 24 = 5^2, 15*(15+1)/2 + 24 = 12^2, 24*(24+1)/2 + 24 = 18^2, 94*(94+1)/2 + 24 = 67^2.
Links
- F. T. Adams-Watters, SeqFan Discussion, Oct 2009
Programs
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Mathematica
Select[Range[12602*10^4],IntegerQ[Sqrt[24+(#(#+1))/2]]&] (* Harvey P. Dale, Jul 07 2019 *)
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PARI
{for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 24), print1(n, ", ") ) );}
Formula
{k: 24+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*(-1-14*x-3*x^2+14*x^3+2*x^4)/((x-1) * (x^2-2*x-1) * (x^2+2*x-1)).
G.f.: ( 4 + 1/(x-1) + (10+27*x)/(x^2-2*x-1) + (-7+4*x)/(x^2+2*x-1) )/2. (End)