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%I #18 May 09 2014 02:49:08
%S 0,1,2,4,9,10,13,16,18,20,25,29,34,36,37,45,49,58,61,64,65,72,73,81,
%T 90,97,100,101,106,121,130,136,137,144,146,148,153,157,160,164,169,
%U 181,193,196,200,202,205,208,218,225,226,232,234,241,244,245
%N Integers that are a sum of two triangular numbers and also the sum of two square numbers (including zeros).
%C This sequence is infinite as, for example, all integers of the form m^8+m^4-2*m^2*n^2+12*m^6*n^2+n^4+38*m^4*n^4+12*m^2*n^6+n^8 are included.
%C The sequence includes all squares, since n^2 = T(n-1) + T(n), where T(n) = A000217(n) is the n-th triangular number. - _Franklin T. Adams-Watters_, Jun 24 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">4425</a>, The American Mathematical Monthly, Vol. 59, No. 6, (June - July 1952), pp. 417-419.
%e 9 is the sum of two triangular numbers: 6 + 3, and also two squares: 9 + 0. Hence 9 is 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[0,250];Prepend[DeleteCases[Table[If[data[[k]]>0,k-1,0],{k,1,Length[data]}],0],0]
%t With[ {n = 250}, Pick[ Range[ 0, n], {} != FindInstance[ a*a + b*b == # && c (c + 1) + d (d + 1) == 2 # && a >= 0 && b >= 0 && c >= 0 && d >= 0, {a, b, c, d}, Integers] & /@ Range[ 0, n]]] (* _Michael Somos_, Jun 24 2011 *)
%Y Cf. A000217, A000290, A191765, intersection of A001481 and A020756.
%K nonn
%O 1,3
%A _Ant King_, Jun 22 2011
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