A230312 -id:A230312 - cp's OEIS frontend

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

Altug Alkan, Jul 06 2016

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.

			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.

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

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 *)

cp's OEIS Frontend

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

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