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

1,1

Links

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

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

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)

Example

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.

Mathematica

Select[Range[68*10^6], IntegerQ[Sqrt[21+(#(#+1))/2]]&] (* Harvey P. Dale, Mar 07 2017 *)

Prog

(PARI) {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 21), print1(n, ", ") ) ); }

Crossrefs

Cf. A000217, A000290, A006451.

Sequence in context: A244260 A178428 A147760 * A340818 A242241 A257745

Adjacent sequences: A154145 A154146 A154147 * A154149 A154150 A154151

Keyword

nonn,less

Author

R. J. Mathar, Oct 18 2009

Status

approved