A103589 -id:A103589

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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

A103582

Binary array below read by downward antidiagonals.

+10
8

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

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

OFFSET

0,1

COMMENTS

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, ...

...

LINKS

Table of n, a(n) for n=0..119.

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.

MATHEMATICA

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 *)

CROSSREFS

Cf. A103583, A103581, A103588, A103589.

KEYWORD

nonn,easy,tabl

AUTHOR

Philippe Deléham, Mar 24 2005

EXTENSIONS

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

Rechecked by David Applegate, Apr 19 2005

STATUS

approved

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.

+10
6

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

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

OFFSET

1,3

COMMENTS

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

LINKS

Table of n, a(n) for n=1..105.

FORMULA

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, ...},

EXAMPLE

Binary expansions of k*2^(k-1), with bits in ascending order by powers of 2, are:

001

0011

000001

0000101

00000011

000000111

00000000001

000000001001

0000000000101

00000000001101

000000000000011

0000000000001011

.................

Giving column sums:

10211132203222433...

MATHEMATICA

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 *)

PROG

(PARI) /* Prints initial 1000 terms: */

{A=vector(1000); for(n=1, #A, Bn=binary(n*2^(n-1)); for(k=1, min(#Bn, #A), A[k]=A[k]+Bn[#Bn-k+1]) ); print(A)}

CROSSREFS

Cf. A089399, A089400, A089401.

KEYWORD

base,nonn

AUTHOR

Paul D. Hanna, Oct 30 2003

STATUS

approved

A103583

Same as A103582, but read antidiagonals in upward direction.

+10
6

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

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

OFFSET

0,1

COMMENTS

Successive digits of A103581.

LINKS

Table of n, a(n) for n=0..122.

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.

MATHEMATICA

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 *)

CROSSREFS

Cf. A103582, A103581, A103588, A103589.

KEYWORD

nonn,easy,tabl

AUTHOR

Philippe Deléham, Mar 24 2005

EXTENSIONS

Rechecked by David Applegate, Apr 19 2005

STATUS

approved

A105033

Read binary numbers downwards to the right.

+10
6

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

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

OFFSET

0,4

COMMENTS

Equals A103530(n+2) - 1. - Philippe Deléham, Apr 06 2005

LINKS

Table of n, a(n) for n=0..75.

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.

FORMULA

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

EXAMPLE

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, ...

MAPLE

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;

MATHEMATICA

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 *)

CROSSREFS

Analog of A102370. Cf. A105034, A105025.

Cf. triangular array in A103589.

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, Apr 04 2005

STATUS

approved

A103588

1's complement of A103582.

+10
5

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

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

OFFSET

0,1

COMMENTS

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, 0, ...)

1 : (1, 0, 0, 0, 0, 0, 0, ...)

2 : (0, 1, 0, 0, 0, 0, 0, ...)

3 : (1, 1, 0, 0, 0, 0, 0, ...)

4 : (0, 0, 1, 0, 0, 0, 0, ...)

5 : (1, 0, 1, 0, 0, 0, 0, ...)

6 : (0, 1, 1, 0, 0, 0, 0, ...)

7 : (1, 1, 1, 0, 0, 0, 0, ...)

... (End)

LINKS

Table of n, a(n) for n=0..104.

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

EXAMPLE

Triangle begins:

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

CROSSREFS

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

KEYWORD

nonn,easy,tabl

AUTHOR

Philippe Deléham, Mar 24 2005

EXTENSIONS

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.

STATUS

approved

A105034

Binary equivalents of A105033.

+10
2

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

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

OFFSET

0,4

COMMENTS

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

LINKS

Table of n, a(n) for n=0..40.

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. A105033, A102370.

Cf. triangular array in A103589.

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, Apr 04 2005

EXTENSIONS

More terms from Benoit Cloitre, Apr 04 2005

STATUS

approved

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