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%I #10 Oct 13 2019 23:43:21
%S 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,
%T 31,60,74,45,1,3,7,15,31,63,108,110,56
%N Successive sequences whose numbers and their differences of increasing rank include all numbers (generalization of A005228).
%C 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).
%C 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,...).
%e 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:
%e 1 3 7 12 18 26 35 45 56 69 83
%e 1 3 7 15 28 47 74 110 156 213 282 ...
%e 1 3 7 15 31 60 108 183 294 451 665 ...
%e 1 3 7 15 31 63 124 233 417 712 1164 ...
%e 1 3 7 15 31 63 127 252 486 904 1617 ...
%e Sequence consists of the terms of this array read by antidiagonals.
%e Of course, the first row is A005228.
%Y Cf. A005228.
%K easy,nonn,uned
%O 0,3
%A Philippe Lallouet (philip.lallouet(AT)orange.fr), Jun 05 2008
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