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

0,2

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.

Links

Table of n, a(n) for n=0..74.

Eric Weisstein's World of Mathematics, Nonarithmetic Progression Sequence.

Index entries related to non-averaging sequences

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.

Example

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, ...
  ...

Crossrefs

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

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

Sequence in context: A332332 A335259 A378037 * A344767 A187883 A134543

Adjacent sequences: A093679 A093680 A093681 * A093683 A093684 A093685

Keyword

nonn,tabl

Author

Ralf Stephan, Apr 09 2004

Status

approved