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%I #16 Apr 29 2016 09:27:17
%S 2,13,18,20,25,29,34,37,58,61,65,72,73,90,97,100,101,106,130,136,137,
%T 146,148,157,160,164,169,181,193,200,202,205,208,218,225,226,232,234,
%U 241,244,245,265,272,274,277,281,288,289,298,306,328,340,346,353,370,373,388,389,400
%N Integers that are a sum of two nonzero triangular numbers and also the sum of two nonzero square numbers.
%C A134422 is a subsequence. - _Franklin T. Adams-Watters_, Jun 25 2011
%H P. A. Piza, <a href="http://www.jstor.org/stable/2308386">Problems for Solution: 4425</a>, The American Mathematical Monthly, Vol. 58, No. 2, (February 1951), p. 113.
%H P. A. Piza, G. W. Walker and C. M. Sandwick, Sr, <a href="http://www.jstor.org/stable/2306829">Problem 4425</a>, The American Mathematical Monthly, Vol. 59, No. 6, (June - July 1952), pp. 417-419.
%e 25 is the sum of two nonzero triangular numbers: 10 + 15, and of two nonzero squares: 9 + 16; so 25 is in the sequence.
%e 9 is the sum of two nonzero triangular numbers: 3 + 6, but can be represented as the sum of two squares only using zero: 0 + 9; so 9 is not in the sequence.
%t data=Length[Reduce[a^2+b^2==1/2 c (c+1)+1/2 d(d+1)== # && a>0 && b>0 && c>0 && d>0,{a,b,c,d},Integers]] &/@Range[400];DeleteCases[Table[If[data[[k]]>0,k,0],{k,1,Length[data]}],0]
%Y Cf. A000217, A000290, A191766, intersection of A000404 and A051533, A134422.
%K nonn,easy
%O 1,1
%A _Ant King_, Jun 22 2011
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