A053603 - OEIS

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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

(list; graph; refs; listen; history; text; internal format)

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.