A099627 -id:A099627 - cp's OEIS frontend

A167979 Linearize the arrays A099627 A124922 ... defined in A167204 and based on A161924 then concatenate to form a new table.

1, 2, 6, 3, 10, 12, 4, 13, 20, 14, 5, 18, 25, 22, 24, 7, 21, 36, 29, 40, 26, 8, 27, 41, 38, 49, 42, 28, 9, 34, 51, 45, 72, 53, 44, 30, 11, 37, 68, 59, 81, 74, 57, 46, 48, 15, 43, 73, 70, 99, 85, 76, 61, 80, 50, 16, 55, 83, 77, 136, 107, 89, 78, 97, 82, 52

Offset: 1

Alford Arnold, Nov 15 2009

Contribution from Alford Arnold, Nov 29 2009: (Start)
Note that the values within A167977 identify the number partitioned described in A125106 and A161924.
(End)

			The resulting table begins:
..1..2..3..4..5..7..8
..6.10.13.18.21.27
.12.20.25.36.41
.14.22.29.38
etc.
Contribution from _Alford Arnold_, Nov 29 2009: 4 equals 2+2 which maps to the natural number 6 (binary 110) and 6 appears in the second array (A124922).

Cf. A099627 A124922 A167204 A161924
Contribution from Alford Arnold, Nov 29 2009: (Start)
A125106(Describes the mapping to partitions). A167977 is A161511(A167979).
(End)

Corrected By Alford Arnold, Nov 29 2009

A176575 Second edge diagonal of table A176577. (The first edge diagonal is A099627).

Original entry on oeis.org

1, 10, 36, 42, 136, 146, 170, 292, 528, 546, 586, 682, 1092, 1170, 2080, 2114, 2184, 2186, 2340, 2346, 2730, 4228, 4370, 4706, 8256, 8322, 8456, 8458, 8738, 8740, 8746, 9362, 9386, 10922, 16644, 16912, 16914, 17476, 17482, 18724, 18730, 32896, 33026

Offset: 1

Alford Arnold, May 13 2010

Sequence A176575 can be useful in reconstructing table A176577.
Consider, for example, diagonal 10 18 21 34 37 43 66 69 75 ...
the highest power of two less than 10 is 8 and 10-8 is 2 (the "residual").
Construct the sequence 10,18,34,66,... by doubling each term and subtracting
the residual. The remaining terms are formed by using the rule "2x+1":
10..18..34..66..
21..37..69..
43..75..
87..

			A176577 begins
1
2...10
3...18...36
4...21...68...42
5...34...73...74..136
7...37..132...85..264..146
so a(n) begins:
1..10..36..42..136..146..

Cf. A099627 A176577

A187769 Triangle read by rows: equivalence classes of natural numbers, where numbers are equivalent when having equal numbers of zeros and ones in binary representation, respectively.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 14, 15, 16, 17, 18, 20, 24, 19, 21, 22, 25, 26, 28, 23, 27, 29, 30, 31, 32, 33, 34, 36, 40, 48, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 47, 55, 59, 61, 62, 63, 64, 65, 66

Offset: 0

Reinhard Zumkeller, Jan 05 2013

Row lengths are given by Pascal's triangle (cf. A007318), seen as flattened sequence, or for n > 0: length of n-th row = A007318(A003056(n-1),A002262(n-1));
1 <= i < j <= length of n-th row: A023416(T(n,i)) = A023416(T(n,j)), A000120(T(n,i)) = A000120(T(n,j)) and A070939(T(n,i)) = A070939(T(n,j));
the table provides a permutation of the natural numbers when seen as flattened sequence.
This sequence can be seen as an irregular triangle S(i,k) where row 0 = {1}, row n = { m = 2^(n-1)..2^n - 1 } sorted according to omega(A019565(m)), where omega = A001221. Under this arrangement, the rows can be further subdivided into segments of m with the same omega(m), which align with the original definition's triangle T. - Michael De Vlieger, Jan 03 2025

			See link.

Reinhard Zumkeller, Rows n = 0..78 of triangle, flattened - all terms < 2^12
Reinhard Zumkeller, Illustration of initial terms
Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^21-1.
Michael De Vlieger, Fan style binary tree of a(n), n = 0..8192, i.e., rows 0..12, with a color function associated with (a(n) mod 2) / 2^floor(log_2 n) that illustrates the relationship with Pascal's triangle.
Index entries for sequences related to binary expansion of n
Index entries for triangles and arrays related to Pascal's triangle
Index entries for sequences that are permutations of the nonnegative integers

Rows of A187786, duplicates removed;
Cf. A099627 (left edge), A023758 (right edge).
Cf. A019565, A294648, A344084.

import List (elemIndices)
a187769 n k = a187769_tabf !! n !! k
a187769_row n = a187769_tabf !! n
a187769_tabf = [0] : [elemIndices (b, len - b) $
   takeWhile ((<= len) . uncurry (+)) $ zip a000120_list a023416_list |
   len <- [1 ..], b <- [1 .. len]]
a187769_list = concat a187769_tabf

{{0}}~Join~Table[SortBy[Range[2^n, 2^(n + 1) - 1], DigitCount[#, 2, 1] &], {n, 0, 8}] // Flatten (* Michael De Vlieger, Jan 03 2025 *)

A073137 a(n) is the least number whose binary representation has the same number of 0's and 1's as n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 5, 7, 8, 9, 9, 11, 9, 11, 11, 15, 16, 17, 17, 19, 17, 19, 19, 23, 17, 19, 19, 23, 19, 23, 23, 31, 32, 33, 33, 35, 33, 35, 35, 39, 33, 35, 35, 39, 35, 39, 39, 47, 33, 35, 35, 39, 35, 39, 39, 47, 35, 39, 39, 47, 39, 47, 47, 63, 64, 65, 65, 67, 65, 67, 67, 71, 65

Offset: 0

Reinhard Zumkeller, Jul 16 2002

nonn

A023416(a(n)) = A023416(n), A000120(a(n)) = A000120(n).

Fixed points are { 0 } union { A099627 }. - Alois P. Heinz, Jan 30 2025

			a(20)=17, as 20='10100' and 17 is the smallest number having two 1's and three 0's: 17='10001', 18='10010', 20='10100' and 24='11000'.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Cf. A007088, A073138, A000523, A073139, A073140, A073141, A072376, A099627.

a:= n-> (l-> (2^nops(l)+2^add(i, i=l))/2-1)(Bits[Split](n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Jun 26 2021

lnb[n_]:=Module[{sidn=Sort[IntegerDigits[n,2]]},FromDigits[Join[{1}, Most[ sidn]],2]]; Join[{0},Array[lnb,80]] (* Harvey P. Dale, Aug 04 2014 *)

a(n) = if(n==0,0, 1<Kevin Ryde, Jun 26 2021

def a(n):
    b = bin(n)[2:]; z = b.count('0'); w = len(b) - z
    return int('1'*(w > 0) + '0'*z + '1'*(w-1), 2)
print([a(n) for n in range(73)]) # Michael S. Branicky, Jun 26 2021

def a(n): b = bin(n)[2:]; return int(b[0] + "".join(sorted(b[1:])), 2)
print([a(n) for n in range(73)]) # Michael S. Branicky, Jun 26 2021

a(0)=0, a(1)=1; for n > 1, let C = 2^(floor(log_2(n))-1) = A072376(n); then a(n) = a(n-C) + C if n < 3*C; otherwise a(n) = 2*a(n - 2*C) + 1. [corrected by Jon E. Schoenfield, Jun 27 2021]

For n > 0: a(n) = (2^(A000120(n) - 1)) * (2^A023416(n) + 1) - 1. - Corrected by Michel Marcus, Nov 15 2013

A324495 Average number of steps t(n) required to get n by repeatedly toggling one of the ceiling(log_2(n)) bits of the binary result of the previous step at a random position with equal probability of the bit positions, starting with all bits 0. The fractional part of t is given separately, i.e., t(n) = a(n) + A324496(n)/A324497(n).

Original entry on oeis.org

1, 3, 4, 7, 9, 9, 10, 15, 18, 18, 20, 18, 20, 20, 21, 31, 37, 37, 40, 37, 40, 40, 41, 37, 40, 40, 41, 40, 41, 41, 42, 63, 74, 74, 78, 74, 78, 78, 80, 74, 78, 78, 80, 78, 80, 80, 82, 74, 78, 78, 80, 78, 80, 80, 82, 78, 80, 80, 82, 80, 82, 82, 83, 127, 147, 147, 153

Offset: 1

Hugo Pfoertner, Mar 05 2019

nonn

The problem is related to random walks on the edges of n-dimensional hypercubes.

a(n) is only dependent on the length of the binary representation A070939(n) and on the binary weight A000120(n).

			a(5) = 9 is given by the sum of occurrence probabilities of toggle chains of even lengths 2*k, multiplied by the lengths.
a(5) = Sum_{k>=1} 4*k*7^(k-1) / 3^(2*k) = 9.
The corresponding simulation results for 10^10 toggle chains are
  2*k Probability P    2*k*P      Cumulated
    2   0.22222334  0.44444668    0.444447
    4   0.17284183  0.69136731    1.135814
    6   0.13442963  0.80657780    1.942392
    8   0.10455718  0.83645746    2.778849
   10   0.08131600  0.81315998    3.592009
   ...
  196   0.00000000  0.00000002    9.000068
.
a(7) = Sum_{k>=1} 2*(2*k+1)*7^(k-1) / 3^(2*k) = 10.

Hugo Pfoertner, Table of n, a(n) for n = 1..4096
P. Diaconis, R. L. Graham, J. A. Morrison, Asymptotic Analysis of a Random Walk on a Hypercube with Many Dimensions, Technical Report EFS NFS 307, Department of Statistics, Stanford University, December 1988.
Persi Diaconis, R. L. Graham, J. A. Morrison, Asymptotic analysis of a random walk on a hypercube with many dimensions", Random Structures & Algorithms, Volume 1, Issue 1, Pages 51-72, Spring 1990.
IBM Research, Ponder This April 2006 - Challenge, Random walks on an n-dimensional hypercube.
IBM Research, Ponder This April 2006 - Challenge, Solution.

Cf. A000120, A003149, A070939, A099627, A324496, A324497.

A099628 Numbers m where m-th Catalan number A000108(m) = binomial(2m,m)/(m+1) is divisible by 2 but not by 4, i.e., where A048881(m) = 1.

Original entry on oeis.org

2, 4, 5, 8, 9, 11, 16, 17, 19, 23, 32, 33, 35, 39, 47, 64, 65, 67, 71, 79, 95, 128, 129, 131, 135, 143, 159, 191, 256, 257, 259, 263, 271, 287, 319, 383, 512, 513, 515, 519, 527, 543, 575, 639, 767, 1024, 1025, 1027, 1031, 1039, 1055, 1087, 1151, 1279, 1535, 2048

Offset: 1

Henry Bottomley, Oct 25 2004

Also, there is exactly one digit position in which both a(n)+1 and a(n)-1, written in binary, have a 1; i.e., the bitwise AND of a(n)-1 and a(n)+1 is 2^k, with k > 0. - Wouter Meeussen, Nov 24 2007

			As triangle, rows start
   2;
   4,  5;
   8,  9, 11;
  16, 17, 19, 23;
  32, 33, 35, 39, 47;
  ...
5 is in the sequence since 10!/(5!6!) = 42 is divisible by 2 but not 4;
6 is not in the sequence since 12!/(6!7!) = 132 is divisible by 4;
7 is not in the sequence since 14!/(7!8!) = 429 is not divisible by 2.
From _Michael De Vlieger_, Dec 28 2022: (Start)
Table showing the binary expansion of a(n) for n = 1..15, replacing 0 with "." to accentuate the pattern of bits:
   n  a(n)  a(n)_2
  ----------------
   1    2       1.
   2    4      1..
   3    5      1.1
   4    8     1...
   5    9     1..1
   6   11     1.11
   7   16    1....
   8   17    1...1
   9   19    1..11
  10   23    1.111
  11   32   1.....
  12   33   1....1
  13   35   1...11
  14   39   1..111
  15   47   1.1111 (End)

Michael De Vlieger, Table of n, a(n) for n = 1..11175 (rows 2..150)
Barry Brent, On the constant terms of certain meromorphic modular forms for Hecke groups, arXiv:2212.12515 [math.NT], 2022.
Barry Brent, On the Constant Terms of Certain Laurent Series, Preprints (2023) 2023061164.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..1830.
Michael De Vlieger, Bitmap showing the binary expansion of a(n) n = 1..300 (24 rows), bits arranged from least to most significant from bottom, n increasing toward the right, where black = 1 and white = 0.

Cf. A000108, A048881, A099627.

/* As triangle */ [[2^(n+1) + 2^k - 1: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 27 2017

Select[Range[2048],IntegerQ[Log[2,BitAnd[ #+1,#-1]]]&] (* Wouter Meeussen, Nov 24 2007 *)
Table[2^(n + 1) + 2^k - 1, {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 28 2022 *)
Select[Range[2100],Boole[Divisible[CatalanNumber[#],{2,4}]]=={1,0}&] (* Harvey P. Dale, Jan 31 2024 *)

from itertools import count, islice
def A099628_gen(): # generator of terms
    m = 1
    for n in count(1):
        m *= 2
        r, k = m-1,1
        for _ in range(n):
            yield r+k
            k *= 2
A099628_list = list(islice(A099628_gen(),40)) # Chai Wah Wu, Nov 15 2022

As triangle, T(n,k) = 2^(n+1) + 2^k - 1 = A099627(n+1, k).

Offset changed to 1 by N. J. A. Sloane, Jul 27 2017

A124922 Second in a series of triangular arrays providing index numbers for subsequences of A060351.

Original entry on oeis.org

6, 10, 13, 18, 21, 27, 34, 37, 43, 55, 66, 69, 75, 87, 111, 130, 133, 139, 151, 175, 223

Offset: 1

Alford Arnold, Nov 21 2006

The first triangular array is A099627 which provides index numbers in A060351 for Pascal's Triangle (A007318). This second array provides the index numbers in A060351 for array A059797.

Note that this table and A099627 are sub-arrays of table A161924 which has A000041 entries per row. - Alford Arnold, Oct 19 2009

			A060351(34,37,43,55) = (14,35,35,14) = Row Four of Array A059797.

Cf. A007318, A059797, A060351, A099627, A124921.

Cf. A000041, A161924.

I would like a clearer definition of this and other recent triangles from this author. - N. J. A. Sloane, Nov 22 2006

More terms from Alford Arnold, Oct 19 2009

A197818 Walsh matrix antidiagonals converted to decimal.

Original entry on oeis.org

1, 3, 5, 15, 17, 51, 93, 255, 257, 771, 1453, 3855, 4593, 13299, 23901, 65535, 65537, 196611, 371373, 983055, 1175281, 3394803, 6103645, 16711935, 16908033, 50593539, 95245741, 252706575, 301011441, 871576563, 1566432605, 4294967295

Offset: 0

Tilman Piesk, Oct 18 2011

Infinite Walsh matrix with the negative ones replaced by zeros (negated binary Walsh matrix), the antidiagonals read as binary numbers.

This sequence is similar to A001317 (Sierpinski triangle rows converted to decimal). a(n) = A001317(n) iff n=0 or n is an element of A099627.

			Top left corner of the negated binary Walsh matrix:
1 1 1 1 1 1 1 1
1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0
1 0 0 1 1 0 0 1
1 1 1 1 0 0 0 0
1 0 1 0 0 1 0 1
1 1 0 0 0 0 1 1
1 0 0 1 0 1 1 0
The antidiagonals in binary and decimal are:
         1 =   1
        11 =   3
       101 =   5
      1111 =  15
     10001 =  17
    110011 =  51
   1011101 =  93
  11111111 = 255

Tilman Piesk, Table of n, a(n) for n = 0..1023
Tilman Piesk, Negated binary Walsh matrix of size 256
Tilman Piesk, The antidiagonals shown in a triangular matrix
Wikipedia, Walsh matrix

Cf. A001317, A099627.

N=2^5;  /* a power of 2 */
parity(x)= {
    my(s=1);
    while ( (x>>s),  x=bitxor(x, x>>s); s+=s; );
    return( bitand(x,1) );
}
W = matrix(N,N, i,j, if(parity(bitand(i-1,j-1)),0,1); );
a(n) = sum(k=0,n, 2^k * W[n-k+1,k+1] );
vector(N,n,a(n-1))
/* Joerg Arndt, Mar 27 2013 */

A167201 Third in a series of triangular subarrays of A117506. Previous arrays are Tables A007318 and A059797.

Original entry on oeis.org

5, 14, 21, 28, 70, 56, 48, 162, 216, 120

Offset: 1

Alford Arnold, Nov 02 2009

This subarray is generated from values related to the source partition 3+3. (cf A161924).

			The domain values begin:
12
20..25
36..41..51
68..73..83..103
so based on function A117506, a(n) begins:
5
14..21
28..70..56
48..162..216..120
Note that A117506(22) maps to Partition 3+3
which corresponds to the 12th natural number appearing in A161924.

A007318 A099627 A117506 A161924

A167202 Fourth in a series of triangular subarrays of A117506. Previous arrays are Tables A007318, A059797 and A167201.

Original entry on oeis.org

5, 21, 14, 56, 70, 28, 120, 216, 162, 48

Offset: 1

Alford Arnold, Nov 04 2009

This subarray is generated from values related to the source partition
2+2+2. (cf A161924). Note that A117506(25) maps to Partition 2+2+2
which corresponds to the 14th natural number appearing in A161924.
Note also that Table A167201 is the transpose of Table a(n) and that
partition 3+3 is the conjugate of partition 2+2+2.

			The A161924 domain values begin:
14
22..29
38..45..59
70..77..91..119
so based on function A117506, a(n) begins:
5
21..14
56..70..28
70..77..91..119

A099627 A007318 A124922 A059797 A161924 A167201 A117506

cp's OEIS Frontend

A167979 Linearize the arrays A099627 A124922 ... defined in A167204 and based on A161924 then concatenate to form a new table.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A176575 Second edge diagonal of table A176577. (The first edge diagonal is A099627).

Views

Author

Keywords

Comments

Examples

Crossrefs

A187769 Triangle read by rows: equivalence classes of natural numbers, where numbers are equivalent when having equal numbers of zeros and ones in binary representation, respectively.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

A073137 a(n) is the least number whose binary representation has the same number of 0's and 1's as n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Python

Formula

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A099628 Numbers m where m-th Catalan number A000108(m) = binomial(2m,m)/(m+1) is divisible by 2 but not by 4, i.e., where A048881(m) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Python

Formula

Extensions

A124922 Second in a series of triangular arrays providing index numbers for subsequences of A060351.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A197818 Walsh matrix antidiagonals converted to decimal.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs