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A309332
Number of ways the n-th triangular number T(n) = A000217(n) can be written as the sum of two positive triangular numbers.
3
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
OFFSET
1,21
The order doesn't matter. 21 = 6+15 = 15+6 are not counted as distinct solutions. - N. J. A. Sloane, Feb 22 2020
FORMULA
a(n) > 0 <=> n in { A012132 }.
a(n) = 0 <=> n in { A027861 }.
a(n) = 1 <=> n in { A108769 }.
EXAMPLE
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.
MAPLE
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);
MATHEMATICA
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];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Nov 16 2022, after Alois P. Heinz *)
CROSSREFS
Cf. A000217, A001652, A012132, A027861, A046080 (the same for squares), A053141, A062301 (the same for primes), A108769, A309507.
Sequence in context: A285982 A261727 A234579 * A109362 A085246 A268726
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 01 2019
STATUS
approved

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Last modified September 20 20:25 EDT 2024. Contains 376077 sequences. (Running on oeis4.)