cp's OEIS Frontend

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

%I A105029 #19 May 08 2020 06:10:03
%S A105029 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,
%T A105029 25,28,62,61,56,51,34,33,36,39,38,37,32,43,42,41,44,47,46,45,40,35,50,
%U A105029 49,52,55,54,53,48,59,58,57,60,126,125,120,115,98,65,68,71,70
%N A105029 Write numbers in binary under each other, left justified, read diagonals in downward direction, convert to decimal.
%C A105029 All terms are distinct, but the numbers 2^m - 1 are missing.
%C A105029 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:
%C A105029 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
%C A105029 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
%C A105029 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
%C A105029 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
%C A105029 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
%C A105029 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
%C A105029 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
%C A105029 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
%C A105029 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...
%C A105029 .......
%C A105029 where n-th row consists of binary expansion of n followed by 0's.
%C A105029 a(n) = A105025(n) iff A070939(n) = A103586(n), cf. A214489. - _Reinhard Zumkeller_, Jul 21 2012
%H A105029 Reinhard Zumkeller, <a href="/A105029/b105029.txt">Table of n, a(n) for n = 0..10000</a>
%H A105029 David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [<a href="http://neilsloane.com/doc/slopey.pdf">pdf</a>, <a href="http://neilsloane.com/doc/slopey.ps">ps</a>].
%H A105029 David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Sloane/sloane300.html">Sloping binary numbers: a new sequence related to the binary numbers</a>, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.
%H A105029 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%e A105029 0
%e A105029 1
%e A105029 1 0
%e A105029 1 1
%e A105029 1 0 0
%e A105029 1 0 1
%e A105029 1 1 0
%e A105029 1 1 1
%e A105029 1 0 0 0
%e A105029 1 0 0 1
%e A105029 1 0 1 0
%e A105029 and reading the diagonals downwards we get 0, 10, 110, 101, 100, 1110, 1101, etc.
%o A105029 (Haskell)
%o A105029 import Data.Bits ((.|.), (.&.))
%o A105029 a105029 n = foldl (.|.) 0 $ zipWith (.&.) a000079_list $
%o A105029    map (\x -> (len + 1 - a070939 x) * x)
%o A105029        (reverse $ enumFromTo n (n - 1 + len))  where len = a103586 n
%o A105029 -- _Reinhard Zumkeller_, Jul 21 2012
%Y A105029 Cf. A102370, A105030, A105025, A105026, A105027, A105028, A105033.
%Y A105029 Cf. A000079, A070939, A103586.
%K A105029 nonn,base
%O A105029 0,2
%A A105029 _Benoit Cloitre_, Apr 03 2005