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A139123
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Successive sequences whose numbers and their differences of increasing rank include all numbers (generalization of A005228).
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0
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1, 1, 3, 1, 3, 7, 12, 1, 3, 7, 15, 18, 1, 3, 7, 15, 28, 26, 1, 3, 7, 15, 31, 47, 35, 1, 3, 7, 15, 31, 60, 74, 45, 1, 3, 7, 15, 31, 63, 108, 110, 56
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OFFSET
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0,3
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COMMENTS
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The condition 2) imposes, for any k, 2 and 4 for values of the first two k-th differences and hence 2^1 to 2^(1+k) for the (1+k) first differences and finally (2^n)-1 for values of k(n){n;1;k+2).
In conclusion, in the limit, the terms of the sequence r(k) will be when k tends to infinity = inf(n) = 2^n - 1 (1,3,7,15,31,63,127,255,511,...).
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LINKS
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EXAMPLE
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Construct the following array where the sequence k(n) of the k-th row is the unique one 1) whose numbers and their k-th differences include exactly all numbers once 2) where both of the sequence and the sequence of their k-th differences are increasing:
1 3 7 12 18 26 35 45 56 69 83
1 3 7 15 28 47 74 110 156 213 282 ...
1 3 7 15 31 60 108 183 294 451 665 ...
1 3 7 15 31 63 124 233 417 712 1164 ...
1 3 7 15 31 63 127 252 486 904 1617 ...
Sequence consists of the terms of this array read by antidiagonals.
Of course, the first row is A005228.
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CROSSREFS
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KEYWORD
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easy,nonn,uned
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AUTHOR
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Philippe Lallouet (philip.lallouet(AT)orange.fr), Jun 05 2008
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STATUS
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approved
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