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

1,3

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

Table of n, a(n) for n=1..56.

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: A365343 A002258 A252760 * A287518 A287526 A287413

Adjacent sequences: A191763 A191764 A191765 * A191767 A191768 A191769

Keyword

nonn

Author

Ant King, Jun 22 2011

Status

approved