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

%I A093682 #22 Feb 16 2025 08:32:53
%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,
%U A093682 56,82,1,31,30,28,20,22,28,32,58,83,163,1,32,31,31,28,23,29,37,59,85
%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 formulas have been proved, however.
%C A093682 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.
%H A093682 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NonarithmeticProgressionSequence.html">Nonarithmetic Progression Sequence.</a>
%H A093682 <a href="/index/No#non_averaging">Index entries related to non-averaging sequences</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^floor((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 Array begins:
%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, ...
%e A093682   ...
%Y A093682 Rows 0-6 are A003278, A004793, A033157, A093678, A093679, A093680, A093681.
%Y A093682 Column 2 is 1+A038754. Cf. A092482, A033158.
%K A093682 nonn,tabl
%O A093682 0,2
%A A093682 _Ralf Stephan_, Apr 09 2004