A139123 Successive sequences whose numbers and their differences of increasing rank include all numbers (generalization of A005228).

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

Offset

0,3

Comments

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,...).

Links

Table of n, a(n) for n=0..41.

Example

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.

Crossrefs

Cf. A005228.

Sequence in context: A086401 A095732 A001644 * A133580 A019603 A171843

Adjacent sequences: A139120 A139121 A139122 * A139124 A139125 A139126

Keyword

easy,nonn,uned

Author

Philippe Lallouet (philip.lallouet(AT)orange.fr), Jun 05 2008

Status

approved