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, 2016, 2080, 2145, 2211, 2415, 2485

Offset: 1

Author

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

Keywords

• nonn
• easy

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

• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,2,-2,0,0,0,0,-1,1).

Programs

Mathematica

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

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

Extensions

More terms from Michel Marcus, May 26 2025