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Search: id:A093682
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| A093682 |
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Array T(m,n) by antidiagonals: nonarithmetic-3-progression sequences with simple closed forms. |
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8
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| 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; listen)
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OFFSET
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0,2
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COMMENT
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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.
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.
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LINKS
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Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
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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).
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.
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EXAMPLE
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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 ...
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CROSSREFS
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Rows 0-6 are A003278, A004793, A033157, A093678, A093679, A093680, A093681.
Column 2 is 1+A038754. Cf. A092482, A033158.
Sequence in context: A132280 A059970 A112157 this_sequence A134543 A093010 A093966
Adjacent sequences: A093679 A093680 A093681 this_sequence A093683 A093684 A093685
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KEYWORD
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nonn,tabl
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AUTHOR
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Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 09 2004
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