A105032 -id:A105032

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A102370

"Sloping binary numbers": write numbers in binary under each other (right-justified), read diagonals in upward direction, convert to decimal.

+10
72

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, 44, 39, 34, 33, 32, 35, 38, 37, 36, 47, 42, 41, 40, 43, 46, 45, 60, 55, 50, 49, 48, 51, 54, 53, 52, 63, 58, 57, 56, 59, 126, 93, 76, 71, 66, 65, 64, 67, 70, 69

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

All terms are distinct, but certain terms (see A102371) are missing. But see A103122.

Trajectory of 1 is 1, 3, 5, 15, 17, 19, 21, 31, 33, ..., see A103192.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)

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. Preprint versions: [pdf, ps].

Index entries for sequences related to binary expansion of n

FORMULA

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

a(n) = A105027(A062289(n)) for n > 0. - Reinhard Zumkeller, Jul 21 2012

EXAMPLE

........0

........1

.......10

.......11

......100

......101

......110

......111

.....1000

.........

The upward-sloping diagonals are:

110

101

100

1111

1010

.......

giving 0, 3, 6, 5, 4, 15, 10, ...

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, ...

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.

MAPLE

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;

MATHEMATICA

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 *)

PROG

(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

(PARI) {a(n) = if( n<1, 0, sum( k=0, length( binary( n)), bitand( n + k, 2^k)))} /* Michael Somos, Mar 26 2012 */

(Haskell)

a102370 n = a102370_list !! n

a102370_list = 0 : map (a105027 . toInteger) a062289_list

-- Reinhard Zumkeller, Jul 21 2012

(Python)

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

CROSSREFS

Related sequences (1): A103542 (binary version), A102371 (complement), A103185, A103528, A103529, A103530, A103318, A034797, A103543, A103581, A103582, A103583.

Related sequences (2): A103584, A103585, A103586, A103587, A103127, A103192 (trajectory of 1), A103122, A103588, A103589, A103202 (sorted), A103205 (base 10 version).

Related sequences (3): A103747 (trajectory of 2), A103621, A103745, A103615, A103842, A103863, A104234, A104235, A103813, A105023, A105024, A105025, A105026, A105027, A105028.

Related sequences (4): A105029, A105030, A105031, A105032, A105033, A105034, A105035, A105108.

Related sequences (5): A105229, A105271, A104378, A104401, A104403, A104489, A104490, A104853, A104893, A104894, A105085.

KEYWORD

nonn,nice,easy,base,look

AUTHOR

Philippe Deléham, Feb 13 2005

EXTENSIONS

More terms from Benoit Cloitre, Mar 20 2005

STATUS

approved

A105031

Binary equivalents of A103185.

+10
2

0, 1, 10, 1, 0, 101, 10, 1, 0, 1, 10, 1, 1000, 101, 10, 1, 0, 1, 10, 1, 0, 101, 10, 1, 0, 1, 10, 10001, 1000, 101, 10, 1, 0, 1, 10, 1, 0, 101, 10, 1, 0, 1, 10, 1, 1000, 101, 10, 1, 0, 1, 10, 1, 0, 101, 10, 1, 0, 1, 100010, 10001, 1000, 101, 10, 1, 0, 1, 10, 1, 0, 101, 10, 1, 0, 1, 10

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..74.

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.

CROSSREFS

Cf. A102370, A104234, A105032.

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, David Applegate, Benoit Cloitre and Philippe Deléham, Apr 03 2005

EXTENSIONS

More terms from Benoit Cloitre, Apr 04 2005

STATUS

approved

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