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A274779 Numbers whose square is the sum of two positive triangular numbers in exactly one way. 0
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 (list; graph; refs; listen; history; text; internal format)
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 A323252

Adjacent sequences:  A274776 A274777 A274778 * A274780 A274781 A274782

KEYWORD

nonn,easy

AUTHOR

Altug Alkan, Jul 06 2016

STATUS

approved

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Last modified July 27 23:16 EDT 2021. Contains 346316 sequences. (Running on oeis4.)