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

1,2

Links

Table of n, a(n) for n=1..20.

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

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)

Example

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.

Mathematica

Select[Range[12602*10^4], IntegerQ[Sqrt[24+(#(#+1))/2]]&] (* Harvey P. Dale, Jul 07 2019 *)

Prog

(PARI) {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 24), print1(n, ", ") ) ); }

Crossrefs

Cf. A000217, A000290, A006451.

Sequence in context: A162964 A067343 A247315 * A324484 A167709 A167353

Adjacent sequences: A154147 A154148 A154149 * A154151 A154152 A154153

Keyword

nonn,less

Author

R. J. Mathar, Oct 18 2009

Status

approved