A191766 Integers that are a sum of two triangular numbers and also the sum of two square numbers (including zeros).

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

Offset: 1

Author

Ant King, Jun 22 2011

Keywords

• nonn

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

Examples

			9 is the sum of two triangular numbers: 6 + 3, and also two squares: 9 + 0. Hence 9 is in the sequence.
		

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.

Crossrefs

Cf. A000217, A000290, A191765, intersection of A001481 and A020756.

Programs

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 *)