A058498 Number of solutions to c(1)t(1) + ... + c(n)t(n) = 0, where c(i) = +-1 for i>1, c(1) = t(1) = 1, t(i) = triangular numbers (A000217).

0, 0, 0, 1, 0, 1, 1, 2, 0, 6, 8, 13, 0, 33, 52, 105, 0, 310, 485, 874, 0, 2974, 5240, 9488, 0, 30418, 55715, 104730, 0, 352467, 642418, 1193879, 0, 4165910, 7762907, 14493951, 0, 50621491, 95133799, 179484713, 0, 637516130, 1202062094, 2273709847, 0, 8173584069

Offset: 1

Author

Naohiro Nomoto, Dec 20 2000

Keywords

• nonn

Examples

			a(8) = 2 because there are two solutions: 1 - 3 + 6 + 10 + 15 - 21 + 28 - 36 = 1 - 3 - 6 + 10 - 15 + 21 + 28 - 36 = 0.
		

Links

• Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 1..633 (first 280 terms from Alois P. Heinz)

Crossrefs

Cf. A000217.

Programs

Maple

b:= proc(n, i) option remember; local m; m:= (2+(3+i)*i)*i/6;
      `if`(n>m, 0, `if`(n=m, 1,
      b(abs(n-i*(i+1)/2), i-1) +b(n+i*(i+1)/2, i-1)))
    end:
a:= n-> `if`(irem(n, 4)=1, 0, b(n*(n+1)/2, n-1)):
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 31 2011

Mathematica

b[n_, i_] := b[n, i] = With[{m = (2+(3+i)*i)*i/6}, If[n>m, 0, If[n == m, 1, b[Abs[n - i*(i+1)/2], i-1] + b[n + i*(i+1)/2, i-1]]]]; a[n_] := If[Mod[n, 4] == 1, 0, b[n*(n+1)/2, n-1]]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jan 30 2017, after Alois P. Heinz *)

Formula

a(n) = [x^(n*(n+1)/2)] Product_{k=1..n-1} (x^(k*(k+1)/2) + 1/x^(k*(k+1)/2)). - Ilya Gutkovskiy, Feb 01 2024

Extensions

More terms from Sascha Kurz, Oct 13 2001

More terms from Alois P. Heinz, Oct 31 2011