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

%I A053603 #19 Jan 06 2024 01:08:54
%S A053603 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,
%T A053603 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,
%U A053603 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
%N A053603 Number of ways to write n as an ordered sum of two nonzero triangular numbers.
%C A053603 a(A051611(n)) = 0; A051533(a(n)) > 0. - _Reinhard Zumkeller_, Jun 27 2013
%H A053603 Reinhard Zumkeller, <a href="/A053603/b053603.txt">Table of n, a(n) for n = 0..10000</a>
%F A053603 G.f.: ( Sum_{k>=1} x^(k*(k+1)/2) )^2. - _Ilya Gutkovskiy_, Dec 24 2016
%F A053603 a(n) = Sum_{k=1..n-1} c(k) * c(n-k), where c(n) = A010054(n). - _Wesley Ivan Hurt_, Jan 06 2024
%t A053603 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 *)
%o A053603 (Haskell)
%o A053603 a053603 n = sum $ map (a010054 . (n -)) $
%o A053603                   takeWhile (< n) $ tail a000217_list
%o A053603 -- _Reinhard Zumkeller_, Jun 27 2013
%o A053603 (PARI)
%o A053603 istriang(n)={n>0 && issquare(8*n+1);}
%o A053603 a(n) = { my(t=1, ct=0, j=1); while (t<n, ct+=istriang(n-t); j+=1; t+=j;); ct; }
%o A053603 \\ _Joerg Arndt_, Sep 05 2013
%Y A053603 Cf. A000217, A007294, A051611, A051533, A052343-A052348, A053604.
%Y A053603 Cf. A010054.
%K A053603 nonn
%O A053603 0,5
%A A053603 _N. J. A. Sloane_, Jan 20 2000