A129966 Triangular numbers which are differences of squares.

0, 1, 3, 15, 21, 28, 36, 45, 55, 91, 105, 120, 136, 153, 171, 231, 253, 276, 300, 325, 351, 435, 465, 496, 528, 561, 595, 703, 741, 780, 820, 861, 903, 1035, 1081, 1128, 1176, 1225, 1275, 1431, 1485, 1540, 1596, 1653, 1711, 1891, 1953

Offset

1,3

Comments

These are the triangular numbers n(n+1)/2 where n is not congruent to 3 or 4 mod 8. (Since the numbers not the difference of two squares are precisely those congruent to 2 mod 4.) - Franklin T. Adams-Watters, Jun 14 2007

Links

Table of n, a(n) for n=1..47.

Formula

G.f. ( -x^2*(x^2+1)*(x^8+2*x^7+11*x^6+4*x^5-4*x^4+4*x^3+11*x^2+2*x+1) ) / ( (1+x)^2*(1+x+x^2)^2*(x^2-x+1)^2*(x-1)^3 ) with a(n) = +a(n-1) +2*a(n-6) -2*a(n-7) -a(n-12) +a(n-13). - R. J. Mathar, Oct 25 2011

Mathematica

With[{n = 100}, Intersection[(#1*((#1 + 1)/2) & ) /@ Range[0, n], Flatten[Outer[ #1^2 - #2^2 &, Range[n], Range[0, n - 1]]]]]

Crossrefs

Sequence in context: A001897 A074214 A036897 * A354675 A243128 A306771

Adjacent sequences: A129963 A129964 A129965 * A129967 A129968 A129969

Keyword

nonn

Author

Peter Pein (petsie(AT)dordos.net), Jun 14 2007

Status

approved