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Triangular numbers which are differences of squares.
1

%I #5 Oct 25 2011 11:07:26

%S 0,1,3,15,21,28,36,45,55,91,105,120,136,153,171,231,253,276,300,325,

%T 351,435,465,496,528,561,595,703,741,780,820,861,903,1035,1081,1128,

%U 1176,1225,1275,1431,1485,1540,1596,1653,1711,1891,1953

%N Triangular numbers which are differences of squares.

%C 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

%F 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

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

%K nonn

%O 1,3

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