A274779 Numbers whose square is the sum of two positive triangular numbers in exactly one way.

2, 3, 5, 6, 7, 8, 10, 12, 13, 18, 20, 27, 28, 33, 37, 42, 45, 47, 55, 58, 60, 62, 63, 65, 67, 73, 75, 78, 80, 85, 88, 90, 92, 102, 103, 105, 112, 115, 118, 120, 125, 128, 130, 132, 135, 140, 142, 150, 153, 157, 163, 170, 175, 192, 193, 198, 200, 203, 210, 215, 218, 220, 222

Offset

1,1

Comments

Obviously, A000217(n) + A000217(n+1) = n*(n+1)/2 + (n+1)*(n+2)/2 = (n+1)^2. So every square that is greater than 1 is the sum of two positive consecutive triangular numbers. This sequence focuses on the squares that have only this trivial solution.

For a related comment, see comments section of A001912.

Links

Table of n, a(n) for n=1..63.

Example

3 is a term because 3^2 is the sum of two positive triangular numbers in exactly 1 way that is: 3^2 = 3 + 6.

Mathematica

nR[n_]:= (SquaresR[2, n]+Plus@@ Pick[{-4, 4}, IntegerQ/@ Sqrt[{n, n/2}]])/8 ; nTr[n_] := nR[8*n + 2] - Boole@ IntegerQ@ Sqrt[8*n + 1]; Select[Range[250], nTr[#^2]==1 &] (* Giovanni Resta, Jul 08 2016 *)

Crossrefs

Cf. A000217, A000290, A001912, A230312, A274758.

Sequence in context: A028790 A028748 A028783 * A120486 A229993 A376592

Adjacent sequences: A274776 A274777 A274778 * A274780 A274781 A274782

Keyword

nonn,easy

Author

Altug Alkan, Jul 06 2016

Status

approved