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%I #41 Dec 07 2019 12:18:24
%S 0,3,6,5,4,15,10,9,8,11,14,13,28,23,18,17,16,19,22,21,20,31,26,25,24,
%T 27,30,61,44,39,34,33,32,35,38,37,36,47,42,41,40,43,46,45,60,55,50,49,
%U 48,51,54,53,52,63,58,57,56,59,126,93,76,71,66,65,64,67,70,69
%N "Sloping binary numbers": write numbers in binary under each other (right-justified), read diagonals in upward direction, convert to decimal.
%C All terms are distinct, but certain terms (see A102371) are missing. But see A103122.
%C Trajectory of 1 is 1, 3, 5, 15, 17, 19, 21, 31, 33, ..., see A103192.
%H Seiichi Manyama, <a href="/A102370/b102370.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)
%H 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. Preprint versions: [<a href="http://neilsloane.com/doc/slopey.pdf">pdf</a>, <a href="http://neilsloane.com/doc/slopey.ps">ps</a>].
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F a(n) = n + Sum_{ k >= 1 such that n + k == 0 mod 2^k } 2^k. (Cf. A103185.) In particular, a(n) >= n. - _N. J. A. Sloane_, Mar 18 2005
%F a(n) = A105027(A062289(n)) for n > 0. - _Reinhard Zumkeller_, Jul 21 2012
%e ........0
%e ........1
%e .......10
%e .......11
%e ......100
%e ......101
%e ......110
%e ......111
%e .....1000
%e .........
%e The upward-sloping diagonals are:
%e 0
%e 11
%e 110
%e 101
%e 100
%e 1111
%e 1010
%e .......
%e giving 0, 3, 6, 5, 4, 15, 10, ...
%e The sequence has a natural decomposition into blocks (see the paper): 0; 3; 6, 5, 4; 15, 10, 9, 8, 11, 14, 13; 28, 23, 18, 17, 16, 19, 22, 21, 20, 31, 26, 25, 24, 27, 30; 61, ...
%e Reading the array of binary numbers along diagonals with slope 1 gives this sequence, slope 2 gives A105085, slope 0 gives A001477 and slope -1 gives A105033.
%p A102370:=proc(n) local t1,l; t1:=n; for l from 1 to n do if n+l mod 2^l = 0 then t1:=t1+2^l; fi; od: t1; end;
%t f[n_] := Block[{k = 1, s = 0, l = Max[2, Floor[Log[2, n + 1] + 2]]}, While[k < l, If[ Mod[n + k, 2^k] == 0, s = s + 2^k]; k++ ]; s]; Table[ f[n] + n, {n, 0, 71}] (* _Robert G. Wilson v_, Mar 21 2005 *)
%o (PARI) A102370(n)=n-1+sum(k=0,ceil(log(n+1)/log(2)),if((n+k)%2^k,0,2^k)) \\ _Benoit Cloitre_, Mar 20 2005
%o (PARI) {a(n) = if( n<1, 0, sum( k=0, length( binary( n)), bitand( n + k, 2^k)))} /* _Michael Somos_, Mar 26 2012 */
%o (Haskell)
%o a102370 n = a102370_list !! n
%o a102370_list = 0 : map (a105027 . toInteger) a062289_list
%o -- _Reinhard Zumkeller_, Jul 21 2012
%o (Python)
%o def a(n): return 0 if n<1 else sum([(n + k)&(2**k) for k in range(len(bin(n)[2:]) + 1)]) # _Indranil Ghosh_, May 03 2017
%Y Related sequences (1): A103542 (binary version), A102371 (complement), A103185, A103528, A103529, A103530, A103318, A034797, A103543, A103581, A103582, A103583.
%Y Related sequences (2): A103584, A103585, A103586, A103587, A103127, A103192 (trajectory of 1), A103122, A103588, A103589, A103202 (sorted), A103205 (base 10 version).
%Y Related sequences (3): A103747 (trajectory of 2), A103621, A103745, A103615, A103842, A103863, A104234, A104235, A103813, A105023, A105024, A105025, A105026, A105027, A105028.
%Y Related sequences (4): A105029, A105030, A105031, A105032, A105033, A105034, A105035, A105108.
%Y Related sequences (5): A105229, A105271, A104378, A104401, A104403, A104489, A104490, A104853, A104893, A104894, A105085.
%K nonn,nice,easy,base,look
%O 0,2
%A _Philippe Deléham_, Feb 13 2005
%E More terms from _Benoit Cloitre_, Mar 20 2005
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