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A309332
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Number of ways the n-th triangular number T(n) = A000217(n) can be written as the sum of two positive triangular numbers.
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3
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0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 2, 0, 3, 0, 0, 1, 1, 3, 0, 0, 1, 0, 1, 0, 0, 3, 1, 1, 0, 1, 3, 0, 1, 1, 1, 2, 0, 1, 2, 0, 1, 1, 2, 1, 1, 1, 1, 2, 1, 0, 3, 1, 1, 1, 0, 3, 1, 1, 0, 0, 2, 0, 1, 1, 1, 1, 1, 5, 0, 1, 1, 0, 1, 0, 0, 3, 0, 3, 1, 0, 3, 1, 3, 1, 3, 3, 0, 1, 0, 0, 3, 0, 2, 0, 1
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OFFSET
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1,21
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COMMENTS
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The order doesn't matter. 21 = 6+15 = 15+6 are not counted as distinct solutions. - N. J. A. Sloane, Feb 22 2020
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LINKS
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FORMULA
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EXAMPLE
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a(3) = 1: 2*3/2 + 2*3/2 = 3*4/2.
a(21) = 2: 6*7/2 + 20*21/2 = 12*13/2 + 17*18/2 = 21*22/2.
a(23) = 3: 9*10/2 + 21*22/2 = 11*12/2 + 20*21/2 = 14*15/2 + 18*19/2 = 23*24/2.
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MAPLE
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a:= proc(n) local h, j, r, w; h, r:= n*(n+1), 0;
for j from n-1 by -1 do w:= j*(j+1);
if 2*w<h then break fi;
if issqr((h-w)*4+1) then r:=r+1 fi
od; r
end:
seq(a(n), n=1..120);
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MATHEMATICA
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a[n_] := Module[{h = n(n+1), j, r = 0, w}, For[j = n-1, True, j--, w = j(j+1); If[2w < h, Break[]]; If[ IntegerQ[Sqrt[4(h-w)+1]], r++]]; r];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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