A093682 Array T(m,n) by antidiagonals: nonarithmetic-3-progression sequences with simple closed forms.

1, 2, 1, 4, 3, 1, 5, 4, 4, 1, 10, 6, 5, 7, 1, 11, 10, 8, 8, 10, 1, 13, 12, 10, 10, 11, 19, 1, 14, 13, 13, 11, 13, 20, 28, 1, 28, 15, 14, 16, 14, 22, 29, 55, 1, 29, 28, 17, 17, 20, 23, 31, 56, 82, 1, 31, 30, 28, 20, 22, 28, 32, 58, 83, 163, 1, 32, 31, 31, 28, 23, 29, 37, 59, 85

Offset: 0

Author

Ralf Stephan, Apr 09 2004

Keywords

• nonn
• tabl

Comments

The nonarithmetic-3-progression sequences starting with a(1)=1, a(2)=1+3^m or 1+2*3^m, m >= 0, seem to have especially simple 'closed' forms. None of these formulas have been proved, however.

T(m,1)=1, T(m,2) = 1 + (1 + [m even])*3^floor(m/2) = 1 + A038754(m), m >= 0, n > 0; T(m,n) is least k such that no three terms of T(m,1), T(m,2), ..., T(m,n-1), k form an arithmetic progression.

Examples

			Array begins:
  1,  2,  4,  5, 10, 11, 13, ...
  1,  3,  4,  6, 10, 12, 13, ...
  1,  4,  5,  8, 10, 13, 14, ...
  1,  7,  8, 10, 11, 16, 17, ...
  1, 10, 11, 13, 14, 20, 22, ...
  ...
		

Links

• Eric Weisstein's World of Mathematics, Nonarithmetic Progression Sequence.
• Index entries related to non-averaging sequences

Crossrefs

Rows 0-6 are A003278, A004793, A033157, A093678, A093679, A093680, A093681.

Column 2 is 1+A038754. Cf. A092482, A033158.

Formula

T(m, n) = (Sum_{k=1..n-1} (3^A007814(k) + 1)/2) + f(n), with f(n) a P-periodic function, where P <= 2^floor((m+3)/2) (conjectured and checked up to m=13, n=1000).

The formula implies that T(m, n) = b(n-1) where b(2n) = 3b(n) + p(n), b(2n+1) = 3b(n) + q(n), with p, q sequences generated by rational o.g.f.s.