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

Author

N. J. A. Sloane, Jan 20 2000

Keywords

• nonn

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

Crossrefs

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

Cf. A010054.

Programs

Haskell

a053603 n = sum $ map (a010054 . (n -)) $
                  takeWhile (< n) $ tail a000217_list
-- Reinhard Zumkeller, Jun 27 2013

Mathematica

nmax = 100; m0 = 10; A053603 := Table[a[n], {n, 0, nmax}]; Clear[counts]; counts[m_] := counts[m] = (Clear[a]; a[A053603);%20counts%5Bm%20=%20m0%5D;%20counts%5Bm%20=%202*m%5D;%20While%5B%20counts%5Bm%5D%20!=%20counts%5Bm/2%5D,%20m%20=%202*m%5D;%20A053603%20(*%20_Jean-Fran%C3%A7ois%20Alcover">] = 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 *)

PARI

istriang(n)={n>0 && issquare(8*n+1);}
a(n) = { my(t=1, ct=0, j=1); while (tJoerg Arndt, Sep 05 2013

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