A053603 Number of ways to write n as an ordered sum of two nonzero triangular numbers.

0, 0, 1, 0, 2, 0, 1, 2, 0, 2, 0, 2, 1, 2, 0, 0, 4, 0, 2, 0, 1, 2, 2, 0, 2, 2, 0, 2, 0, 2, 1, 4, 0, 0, 2, 0, 2, 2, 2, 2, 0, 0, 3, 2, 0, 0, 4, 0, 2, 2, 0, 4, 0, 0, 0, 2, 3, 2, 2, 0, 2, 2, 0, 0, 2, 2, 2, 2, 0, 2, 2, 0, 3, 2, 0, 0, 4, 0, 0, 2, 0, 6, 0, 2, 2, 0, 0, 2, 2, 0, 1, 2, 2

Offset

0,5

Comments

a(A051611(n)) = 0; A051533(a(n)) > 0. - Reinhard Zumkeller, Jun 27 2013

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Formula

G.f.: ( Sum_{k>=1} x^(k*(k+1)/2) )^2. - Ilya Gutkovskiy, Dec 24 2016

a(n) = Sum_{k=1..n-1} c(k) * c(n-k), where c(n) = A010054(n). - Wesley Ivan Hurt, Jan 06 2024

Mathematica

nmax = 100; m0 = 10; A053603 := Table[a[n], {n, 0, nmax}]; Clear[counts]; counts[m_] := counts[m] = (Clear[a]; a[_] = 0; Do[k = i*(i+1)/2 + j*(j+1)/2; a[k] = a[k]+1, {i, 1, m}, {j, 1, m}]; A053603); counts[m = m0]; counts[m = 2*m]; While[ counts[m] != counts[m/2], m = 2*m]; A053603 (* Jean-François Alcover, Sep 05 2013 *)

Prog

(Haskell)
a053603 n = sum $ map (a010054 . (n -)) $
                  takeWhile (< n) $ tail a000217_list
-- Reinhard Zumkeller, Jun 27 2013
(PARI)
istriang(n)={n>0 && issquare(8*n+1); }
a(n) = { my(t=1, ct=0, j=1); while (t<n, ct+=istriang(n-t); j+=1; t+=j; ); ct; }
\\ Joerg Arndt, Sep 05 2013

Crossrefs

Cf. A000217, A007294, A051611, A051533, A052343-A052348, A053604.

Cf. A010054.

Sequence in context: A283310 A331534 A035445 * A371222 A085794 A336939

Adjacent sequences: A053600 A053601 A053602 * A053604 A053605 A053606

Keyword

nonn

Author

N. J. A. Sloane, Jan 20 2000

Status

approved