A309332 Number of ways the n-th triangular number T(n) = A000217(n) can be written as the sum of two positive triangular numbers.

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

Comments

The order doesn't matter. 21 = 6+15 = 15+6 are not counted as distinct solutions. - N. J. A. Sloane, Feb 22 2020

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000

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

Adjacent sequences: A309329 A309330 A309331 * A309333 A309334 A309335

Keyword

nonn

Author

Alois P. Heinz, Aug 01 2019

Status

approved