

A093682


Array T(m,n) by antidiagonals: nonarithmetic3progression sequences with simple closed forms.


15



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



OFFSET

0,2


COMMENTS

The nonarithmetic3progression 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^[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,n1),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 nonaveraging sequences


FORMULA

T(m, n) = sum[k=1, n1, (3^A007814(k)+1)/2] + f(n), with f(n) a Pperiodic function, where P <= 2^[(m+3)/2] (conjectured and checked up to m=13, n=1000).
The formula implies that T(m, n)=b(n1) 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

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 06 are A003278, A004793, A033157, A093678, A093679, A093680, A093681.
Column 2 is 1+A038754. Cf. A092482, A033158.
Sequence in context: A258090 A112157 A265624 * A187883 A134543 A305540
Adjacent sequences: A093679 A093680 A093681 * A093683 A093684 A093685


KEYWORD

nonn,tabl


AUTHOR

Ralf Stephan, Apr 09 2004


STATUS

approved



