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A274779
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Numbers whose square is the sum of two positive triangular numbers in exactly one way.
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0
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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