Search: id:A093682
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%I A093682
%S A093682 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,
%T A093682 13,13,11,13,20,28,1,28,15,14,16,14,22,29,55,1,29,28,17,17,20,23,31,56,
%U A093682 82,1,31,30,28,20,22,28,32,58,83,163,1,32,31,31,28,23,29,37,59,85
%N A093682 Array T(m,n) by antidiagonals: nonarithmetic-3-progression sequences 
               with simple closed forms.
%C A093682 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 formulae have been proved, however.
%C A093682 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,
               n-1),k form an arithmetic progression.
%H A093682 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               NonarithmeticProgressionSequence.html">Link to a section of The World 
               of Mathematics.</a>
%F A093682 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^[(m+3)/2] (conjectured and checked up to m=13, 
               n=1000).
%F A093682 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.
%e A093682 1 2 4 5 10 11 13 ...
%e A093682 1 3 4 6 10 12 13 ...
%e A093682 1 4 5 8 10 13 14 ...
%e A093682 1 7 8 10 11 16 17 ...
%e A093682 1 10 11 13 14 20 22 ...
%Y A093682 Rows 0-6 are A003278, A004793, A033157, A093678, A093679, A093680, A093681.
%Y A093682 Column 2 is 1+A038754. Cf. A092482, A033158.
%Y A093682 Sequence in context: A132280 A059970 A112157 this_sequence A134543 A093010 
               A093966
%Y A093682 Adjacent sequences: A093679 A093680 A093681 this_sequence A093683 A093684 
               A093685
%K A093682 nonn,tabl
%O A093682 0,2
%A A093682 Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 09 2004

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