A105029 Write numbers in binary under each other, left justified, read diagonals in downward direction, convert to decimal.

0, 2, 6, 5, 4, 14, 13, 8, 11, 10, 9, 12, 30, 29, 24, 19, 18, 17, 20, 23, 22, 21, 16, 27, 26, 25, 28, 62, 61, 56, 51, 34, 33, 36, 39, 38, 37, 32, 43, 42, 41, 44, 47, 46, 45, 40, 35, 50, 49, 52, 55, 54, 53, 48, 59, 58, 57, 60, 126, 125, 120, 115, 98, 65, 68, 71, 70

Offset

0,2

Comments

All terms are distinct, but the numbers 2^m - 1 are missing.

a(n) = Sum_{k>=1} B(n+k-1,k)*2^(A103586(n)-k) where B(n,k) n>=1, k>=1 is the infinite array:

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...

1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...

1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...

1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...

.......

where n-th row consists of binary expansion of n followed by 0's.

a(n) = A105025(n) iff A070939(n) = A103586(n), cf. A214489. - Reinhard Zumkeller, Jul 21 2012

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

Index entries for sequences related to binary expansion of n

Example

0
1
1 0
1 1
1 0 0
1 0 1
1 1 0
1 1 1
1 0 0 0
1 0 0 1
1 0 1 0
and reading the diagonals downwards we get 0, 10, 110, 101, 100, 1110, 1101, etc.

Prog

(Haskell)
import Data.Bits ((.|.), (.&.))
a105029 n = foldl (.|.) 0 $ zipWith (.&.) a000079_list $
   map (\x -> (len + 1 - a070939 x) * x)
       (reverse $ enumFromTo n (n - 1 + len))  where len = a103586 n
-- Reinhard Zumkeller, Jul 21 2012

Crossrefs

Cf. A102370, A105030, A105025, A105026, A105027, A105028, A105033.

Cf. A000079, A070939, A103586.

Sequence in context: A372060 A198821 A171897 * A309040 A316134 A273621

Adjacent sequences: A105026 A105027 A105028 * A105030 A105031 A105032

Keyword

nonn,base

Author

Benoit Cloitre, Apr 03 2005

Status

approved