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 A191766 Integers that are a sum of two triangular numbers and also the sum of two square numbers (including zeros). 1
 0, 1, 2, 4, 9, 10, 13, 16, 18, 20, 25, 29, 34, 36, 37, 45, 49, 58, 61, 64, 65, 72, 73, 81, 90, 97, 100, 101, 106, 121, 130, 136, 137, 144, 146, 148, 153, 157, 160, 164, 169, 181, 193, 196, 200, 202, 205, 208, 218, 225, 226, 232, 234, 241, 244, 245 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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. 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 LINKS P. A. Piza, Problems for Solution: 4425 The American Mathematical Monthly, Vol. 58, No. 2, (February 1951), p. 113. P. A. Piza, G. W. Walker, and C. M. Sandwick, Sr., 4425, The American Mathematical Monthly, Vol. 59, No. 6, (June - July 1952), pp. 417-419. EXAMPLE 9 is the sum of two triangular numbers: 6 + 3, and also two squares: 9 + 0. Hence 9 is in the sequence. MATHEMATICA 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] 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 *) CROSSREFS Cf. A000217, A000290, A191765, intersection of A001481 and A020756. Sequence in context: A047465 A002258 A252760 * A287518 A287526 A287413 Adjacent sequences:  A191763 A191764 A191765 * A191767 A191768 A191769 KEYWORD nonn AUTHOR Ant King, Jun 22 2011 STATUS approved

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Last modified June 2 18:15 EDT 2020. Contains 334787 sequences. (Running on oeis4.)