A100922 -id:A100922 - cp's OEIS frontend

A163510 Irregular table read by rows: Write n in binary. For each 1, the m-th term of row n is the number of 0's between the m-th 1, reading right to left, and the (m-1)th 1 (or the right side of the number if m-1 = 0).

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

Offset: 1

Leroy Quet, Jul 29 2009

Row n contains exactly A000120(n) terms, for each n.
All odd-numbered rows begin with 0. All even-numbered rows begin with a positive integer.
Can be used to compute the permutation A163511.

			Table begins as:
  Row  n in    Terms on
   n   binary  that row
   1      1    0; (the distance of 1-bit from the right edge is zero)
   2     10    1; (the distance of 1-bit from the right edge is one)
   3     11    0,0;
   4    100    2;
   5    101    0,1; (the least significant 1-bit is zero steps away from the right edge, and there is one zero between those two 1-bits)
   6    110    1,0;
   7    111    0,0,0;
   8   1000    3;
   9   1001    0,2;
  10   1010    1,1;
  11   1011    0,0,1;
  12   1100    2,0;
  13   1101    0,1,0;
  14   1110    1,0,0;
  15   1111    0,0,0,0;
  16  10000    4;

Antti Karttunen, Table of n, a(n) for n = 1..11265

Cf. A000120, A006068, A100922, A227186, A243067, A163511, A227736.

Equals A228351-1, termwise.

Table[Reverse@ Map[Ceiling[(Length@ # - 1)/2] &, DeleteCases[Split@ Join[Riffle[IntegerDigits[n, 2], 0], {0}], {k__} /; k == 1]], {n, 46}] // Flatten (* Michael De Vlieger, Jul 25 2016 *)

from itertools import count, islice
def A163510_gen(): # generator of terms
    for n in count(1):
        k = n
        while k:
            yield (s:=(~k&k-1).bit_length())
            k >>= s+1
A163510_list = list(islice(A163510_gen(),30)) # Chai Wah Wu, Jul 17 2023

(define (A163510 n) (- (A227186bi (A006068 (A100922 (- n 1))) (A243067 n)) 1))
;; See A227186 for A227186bi. - Antti Karttunen, Jun 19 2014

a(n) = A227186(A006068(A100922(n-1)), A243067(n)) - 1. - Antti Karttunen, Jun 19 2014

Additional terms computed and Example section added by Antti Karttunen, Jun 19 2014

A243067 Integers from 0 to A000120(n)-1 followed by integers from 0 to A000120(n+1)-1 and so on, starting with n=1.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0

Offset: 1

Antti Karttunen, Jun 19 2014

nonn

			For n=1, also 1 in binary notation, so the count of its 1-bits is 1 (A000120(1)=1), we list numbers from 0 to 0, thus just 0.
For n=2, 10 in binary, thus A000120(2)=1, we list numbers from 0 to 0, thus just 0.
For n=3, 11 in binary, thus A000120(3)=2, we list numbers from 0 to 1, and so we have the first four terms of the sequence: 0; 0; 0, 1;

Cf. A000788, A100922, A163510, A163511, A227740.

(define (A243067 n) (- n (+ 1 (A000788 (- (A100922 (- n 1)) 1)))))

a(n) = n - (1 + A000788(A100922(n-1)-1)).

A100921 n appears A023416(n) times (appearances equal number of 0-bits).

Original entry on oeis.org

0, 2, 4, 4, 5, 6, 8, 8, 8, 9, 9, 10, 10, 11, 12, 12, 13, 14, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 23, 24, 24, 24, 25, 25, 26, 26, 27, 28, 28, 29, 30, 32, 32, 32, 32, 32, 33, 33, 33, 33, 34, 34, 34, 34, 35, 35, 35, 36, 36, 36, 36, 37, 37, 37

Offset: 0

Rick L. Shepherd, Nov 21 2004

			The binary representation of 16 is 10000, which has four 0-bits (and one 1-bit), hence 16 appears four times in this sequence (but only once in A100922).

Cf. A100922 (n's appearances equal its number of 1-bits), A030530 (n's appearances equal its total number of bits), A023416, A059009.

Flatten[Table[Table[n, {DigitCount[n, 2, 0]}], {n, 0, 37}]] (* Amiram Eldar, Feb 18 2024 *)

def A059015(n): return 2+(n+1)*((t:=(n+1).bit_length())-n.bit_count())-(1<>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1))>>1)
def A100921(n):
    if n == 0: return 0
    m, k = 1, 1
    while A059015(m)<=n: m<<=1
    while m-k>1:
        r = m+k>>1
        if A059015(r)>n:
            m = r
        else:
            k = r
    return m  # Chai Wah Wu, Nov 11 2024

Sum_{n>=1} (-1)^(n+1)/a(n) = Sum_{n>=1} (-1)^(n+1)/A059009(n) = 0.395592509... . - Amiram Eldar, Feb 18 2024

cp's OEIS Frontend

A163510 Irregular table read by rows: Write n in binary. For each 1, the m-th term of row n is the number of 0's between the m-th 1, reading right to left, and the (m-1)th 1 (or the right side of the number if m-1 = 0).

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A243067 Integers from 0 to A000120(n)-1 followed by integers from 0 to A000120(n+1)-1 and so on, starting with n=1.

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A100921 n appears A023416(n) times (appearances equal number of 0-bits).

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