A103589 -id:A103589 - cp's OEIS frontend

A102370 "Sloping binary numbers": write numbers in binary under each other (right-justified), read diagonals in upward direction, convert to decimal.

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

Offset: 0

Philippe Deléham, Feb 13 2005

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.

			........0
........1
.......10
.......11
......100
......101
......110
......111
.....1000
.........
The upward-sloping diagonals are:
0
11
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.

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

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.

a102370 n = a102370_list !! n
a102370_list = 0 : map (a105027 . toInteger) a062289_list
-- Reinhard Zumkeller, Jul 21 2012

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;

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 *)

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

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

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

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

More terms from Benoit Cloitre, Mar 20 2005

A103582 Binary array below read by downward antidiagonals.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Philippe Deléham, Mar 24 2005

The k-th row has alternating blocks of 2^k 1's followed by 2^k 0's:
1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...
1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, ...
1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, ...
1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
...

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.

Cf. A103583, A103581, A103588, A103589.

t = Table[ Take[ Flatten[ Table[ Join[ Table[1, {i, n}], Table[0, {i, n}]], {10}]], 15], {n, 15}]; Flatten[ Table[ t[[i, n - i + 1]], {n, 14}, {i, n}]] (* Robert G. Wilson v, Mar 24 2005 *)

More terms from Robert G. Wilson v and Benoit Cloitre, Mar 24 2005

Rechecked by David Applegate, Apr 19 2005

A089398 a(n) = n-th column sum of binary digits of k*2^(k-1), where summation is over k>=1, without carrying between columns.

Original entry on oeis.org

1, 0, 2, 1, 1, 1, 3, 2, 2, 0, 3, 2, 2, 2, 4, 3, 3, 1, 2, 2, 2, 2, 4, 3, 3, 1, 4, 3, 3, 3, 5, 4, 4, 2, 3, 1, 2, 2, 4, 3, 3, 1, 4, 3, 3, 3, 5, 4, 4, 2, 3, 3, 3, 3, 5, 4, 4, 2, 5, 4, 4, 4, 6, 5, 5, 3, 4, 2, 1, 2, 4, 3, 3, 1, 4, 3, 3, 3, 5, 4, 4, 2, 3, 3, 3, 3, 5, 4, 4, 2, 5, 4, 4, 4, 6, 5, 5, 3, 4, 2, 3, 3, 5, 4, 4

Offset: 1

Paul D. Hanna, Oct 30 2003

sum(k=1,n, a(k)*2^(k-1)) = 2^A089399(n)+1 for n>2, with a(1)=a(2)=1.
Row sums of triangular arrays in A103588 and in A103589. - Philippe Deléham, Apr 04 2005
a(k) = 0 for k = 2, 10, 2058, 2058 + 2^2059, ..., that is, for k = A034797(n) - 1, n>=2. - Philippe Deléham, Nov 16 2007

			Binary expansions of k*2^(k-1), with bits in ascending order by powers of 2, are:
1
001
0011
000001
0000101
00000011
000000111
00000000001
000000001001
0000000000101
00000000001101
000000000000011
0000000000001011
.................
Giving column sums:
10211132203222433...

Cf. A089399, A089400, A089401.

f[n_] := Block[{lg = Floor[Log[2, n]] + 1}, Sum[ Join[ Reverse[ IntegerDigits[n - i + 1, 2]], {0}][[i]], {i, lg}]]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Mar 26 2005 *)

a(2^n)=n-1 (for n>0), a(2^n-1)=n (for n>0), a(2^n+1)=n-1 (for n>1), a(2^n-k)=n-A089400(k) (for n>k>0), a(2^n+k)=n-A089401(k) (for n>k>0), where sequences have limits: A089400={0, 2, 2, 2, 1, 4, 2, 2, 1, 3, 3, ...} and A089401={1, 1, 3, 2, 4, 5, 6, 5, 7, 8, 11, 9, ...},

A103583 Same as A103582, but read antidiagonals in upward direction.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1

Offset: 0

Philippe Deléham, Mar 24 2005

Successive digits of A103581.

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.

Cf. A103582, A103581, A103588, A103589.

t = Table[ Take[ Flatten[ Table[ Join[ Table[1, {i, n}], Table[0, {i, n}]], {10}]], 15], {n, 15}]; Flatten[ Table[ t[[n - i + 1, i]], {n, 14}, {i, n}]] (* Robert G. Wilson v, Mar 24 2005 *)

Rechecked by David Applegate, Apr 19 2005

A105033 Read binary numbers downwards to the right.

Original entry on oeis.org

0, 1, 0, 3, 2, 1, 4, 7, 6, 5, 0, 11, 10, 9, 12, 15, 14, 13, 8, 3, 18, 17, 20, 23, 22, 21, 16, 27, 26, 25, 28, 31, 30, 29, 24, 19, 2, 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, 63, 62, 61, 56, 51, 34, 1, 68, 71, 70, 69, 64, 75

Offset: 0

N. J. A. Sloane, Apr 04 2005

Equals A103530(n+2) - 1. - Philippe Deléham, Apr 06 2005
This sequence can also be produced as follows:
  Using binary arithmetic, start with zero and repeatedly add 1 while deferring carries one iteration.
    a(0) = 0, c(0) = 1
    a(n) = a(n-1) XOR c(n-1)
    c(n) = (a(n-1) AND c(n-1))*2+1
  where c is the carries, XOR is bitwise exclusive-or, and AND is bitwise and.
This has the property that a(n) = n-c(n)+1. - Christopher Scussel, Jan 31 2025

			Start with the binary numbers:
  ......0
  ......1
  .....10
  .....11
  ....100
  ....101
  ....110
  ....111
  ...1000
  .......
and read downwards to the right, getting 0, 1, 0, 11, 10, 1, 100, 111, ...

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.

Analog of A102370. Cf. A105034, A105025.

Cf. triangular array in A103589.

f:= proc (n) local t1, l; t1 := n; for l from 0 to n do if `mod`(n-l,2^(l+1)) = 0 and n >= 2^(l+1) then t1 := t1-2^(l+1) fi; od; t1; end proc;

f[n_] := Block[{k = 0, s = 0}, While[2^(k + 1) < n + 1, If[ Mod[n, 2^(k + 1)] == k, s = s + 2^(k + 1)]; k++ ]; n - s]; Table[ f[n], {n, 0, 75}] (* Robert G. Wilson v, Apr 06 2005 *)

a(n) = n - Sum_{ k >= 0, 2^{k+1} <= n, n == k mod 2^(k+1) } 2^(k+1).
Structure: blocks of size 2^k taken from A105025, interspersed with terms a(n) itself! Thus a(2^k + k - 1 ) = a(k-1) for k >= 1.
From David Applegate, Apr 06 2005: (Start)
"a(n) = 2^k + a(n-2^k) if k >= 1 and 0 <= n - 2^k - k < 2^k, = a(n-2^k) if k >= 1 and n - 2^k - k = -1, or = 0 if n = 0 (and exactly one of the three conditions is true for any n >= 0).
"Equivalently, a(2^k + k + x) = 2^k + a(k+x) if 0 <= x < 2^k, = a(k+x) if x = -1 (for each n >= 0, there is a unique k, x such that 2^k + k + x = n, k >= 0, -1 <= x < 2^k). This recurrence follows immediately from the definition.
"The recurrence captures three observed facts about a: a(2^k + k - 1) = a(k-1); a consists of blocks of length 2^k of A105025 interspersed with terms of a; a(n) = n - Sum_{ k >= 0, 2^{k+1} <= n, n = k mod 2^(k+1) } 2^(k+1)."  (End)
a(n) = sum_{k=0..n} A103589(n,k)*2^(n-k). - L. Edson Jeffery, Dec 01 2013

A103588 1's complement of A103582.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 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, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 24 2005

Comment from Jared Benjamin Ricks (jaredricks(AT)yahoo.com), Jan 31 2009: (Start)
This sequence be also be obtained in the following way. Write numbers in binary from left to right and read the resulting array by antidiagonals upwards:
: (0, 0, 0, 0, 0, 0, 0, ...)
: (1, 0, 0, 0, 0, 0, 0, ...)
: (0, 1, 0, 0, 0, 0, 0, ...)
: (1, 1, 0, 0, 0, 0, 0, ...)
: (0, 0, 1, 0, 0, 0, 0, ...)
: (1, 0, 1, 0, 0, 0, 0, ...)
: (0, 1, 1, 0, 0, 0, 0, ...)
: (1, 1, 1, 0, 0, 0, 0, ...)
... (End)

			Triangle begins:
0
1 0
0 0 0
1 1 0 0
0 1 0 0 0
1 0 0 0 0 0
0 0 1 0 0 0 0
1 1 1 0 0 0 0 0
0 1 1 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 0 0
1 1 0 1 0 0 0 0 0 0 0 0

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

Cf. A103582, A103581, A103589. Considered as a triangle, obtained by reversing the rows of the triangle in A103589.

More terms from Robert G. Wilson v and Benoit Cloitre, Mar 26 2005
Corrected by N. J. A. Sloane, Apr 19 2005
Rechecked by David Applegate, Apr 19 2005.

A105034 Binary equivalents of A105033.

Original entry on oeis.org

0, 1, 0, 11, 10, 1, 100, 111, 110, 101, 0, 1011, 1010, 1001, 1100, 1111, 1110, 1101, 1000, 11, 10010, 10001, 10100, 10111, 10110, 10101, 10000, 11011, 11010, 11001, 11100, 11111, 11110, 11101, 11000, 10011, 10, 100001, 100100, 100111, 100110

Offset: 0

N. J. A. Sloane, Apr 04 2005

Number of 1's in a(n) is A089398(n). - Philippe Deléham, Apr 05 2005.

The version 0, 01, 000, 0011, 00010, 000001, ... is obtained by interchanging 0 and 1 in A103581: 1, 10, 111, 1100, 11101, 111110, .... - Philippe Deléham, Apr 07 2005

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.

Cf. A105033, A102370.

Cf. triangular array in A103589.

More terms from Benoit Cloitre, Apr 04 2005

cp's OEIS Frontend

A102370 "Sloping binary numbers": write numbers in binary under each other (right-justified), read diagonals in upward direction, convert to decimal.

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A103582 Binary array below read by downward antidiagonals.

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A089398 a(n) = n-th column sum of binary digits of k*2^(k-1), where summation is over k>=1, without carrying between columns.

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A103583 Same as A103582, but read antidiagonals in upward direction.

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A105033 Read binary numbers downwards to the right.

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A103588 1's complement of A103582.

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A105034 Binary equivalents of A105033.

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