%I A102370
%S A102370 0,3,6,5,4,15,10,9,8,11,14,13,28,23,18,17,16,19,22,21,20,31,26,25,
%T A102370 24,27,30,61,44,39,34,33,32,35,38,37,36,47,42,41,40,43,46,45,60,55,
%U A102370 50,49,48,51,54,53,52,63,58,57,56,59,126,93,76,71,66,65,64,67,70,69
%N A102370 "Sloping binary numbers": write numbers in binary under each other (right-justified), 
               read diagonals in upward direction, convert to decimal.
%C A102370 All terms are distinct, but certain terms (see A102371) are missing. 
               But see A103122.
%C A102370 Trajectory of 1 is 1, 3, 5, 15, 17, 19, 21, 31, 33, ..., see A103192.
%D A102370 David Applegate, Benoit Cloitre, Philippe DELEHAM 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.
%H A102370 T. D. Noe, <a href="b102370.txt">Table of n, a(n) for n=0..1000</a>
%H A102370 David Applegate, Benoit Cloitre, Philippe DELEHAM and N. J. A. Sloane, 
               Sloping binary numbers: a new sequence related to the binary numbers 
               [<a href="http://www.research.att.com/~njas/doc/slopey.pdf">pdf</
               a>, <a href="http://www.research.att.com/~njas/doc/slopey.ps">ps</
               a>].
%F A102370 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 (njas(AT)research.att.com), 
               Mar 18, 2005
%e A102370 ........0
%e A102370 ........1
%e A102370 .......10
%e A102370 .......11
%e A102370 ......100
%e A102370 ......101
%e A102370 ......110
%e A102370 ......111
%e A102370 .....1000
%e A102370 .........
%e A102370 The upward-sloping diagonals are:
%e A102370 0
%e A102370 11
%e A102370 110
%e A102370 101
%e A102370 100
%e A102370 1111
%e A102370 1010
%e A102370 .......
%e A102370 giving 0, 3, 6, 5, 4, 15, 10, ...
%e A102370 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 A102370 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 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 A102370 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}] (from Robert G. Wilson v Mar 21 2005)
%o A102370 (PARI) A102370(n)=n-1+sum(k=0,ceil(log(n+1)/log(2)),if((n+k)%2^k,0,2^k)) 
               (Cloitre, Mar 20 2005)
%Y A102370 Related sequences (1): A103542 (binary version), A102371 (complement), 
               A103185, A103528, A103529, A103530, A103318, A034797, A103543, A103581, 
               A103582, A103583.
%Y A102370 Related sequences (2): A103584, A103585, A103586, A103587, A103127, A103192 
               (trajectory of 1), A103122, A103588, A103589, A103202 (sorted), A103205 
               (base 10 version).
%Y A102370 Related sequences (3): A103747 (trajectory of 2), A103621, A103745, A103615, 
               A103842, A103863, A104234, A104235, A103813, A105023, A105024, A105025, 
               A105026, A105027, A105028.
%Y A102370 Related sequences (4): A105029, A105030, A105031, A105032, A105033, A105034, 
               A105035, A105108.
%Y A102370 Related sequences (5): A105229, A105271, A104378, A104401, A104403, A104489, 
               A104490, A104853, A104893, A104894, A105085.
%Y A102370 Sequence in context: A159057 A159067 A159058 this_sequence A106109 A123688 
               A082284
%Y A102370 Adjacent sequences: A102367 A102368 A102369 this_sequence A102371 A102372 
               A102373
%K A102370 nonn,nice,easy
%O A102370 0,2
%A A102370 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 13 2005
%E A102370 More terms from Benoit Cloitre, Mar 20 2005