A079071 -id:A079071 - cp's OEIS frontend

A079074 Sum of numbers < n having in binary representation the same number of 0's and 1's as n.

0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 9, 0, 19, 11, 24, 0, 0, 0, 17, 0, 35, 19, 40, 0, 55, 62, 87, 23, 113, 50, 79, 0, 0, 0, 33, 0, 67, 35, 72, 0, 103, 110, 151, 39, 193, 82, 127, 0, 143, 237, 286, 173, 336, 224, 277, 47, 388, 331, 388, 102, 446, 161, 222, 0, 0, 0, 65, 0, 131, 67, 136, 0

Offset: 0

Reinhard Zumkeller, Dec 21 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..16383

Cf. A079072, A079073, A079071, A007088, A023416, A000120.

f:= n-> (x-> (t-> t*(t+1)/2+x[2])(x[1]+x[2]))(add(
    `if`(i=0, [1, 0], [0, 1]), i=convert(n, base, 2))):
b:= proc(n) b(n):= b(n-1)+n*x^f(n) end: b(-1):=0:
a:= n-> coeff(b(n-1), x, f(n)):
seq(a(n), n=0..150);  # Alois P. Heinz, Feb 08 2018

bdQ[m_,n_]:=Module[{a=DigitCount[m,2,0],b=DigitCount[m,2,1], c= DigitCount[ n,2,0], d=DigitCount[ n,2,1]}, a==c&&b==d]; Table[Total[ Select[Range[n-1],bdQ[#,n]&]],{n,80}] (* Harvey P. Dale, Sep 08 2011 *)

A079070 Number of numbers < n having in binary representation the same number of 0's as n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 2, 0, 1, 2, 3, 3, 4, 5, 3, 0, 1, 2, 4, 3, 5, 6, 6, 4, 7, 8, 7, 9, 8, 9, 4, 0, 1, 2, 5, 3, 6, 7, 10, 4, 8, 9, 11, 10, 12, 13, 10, 5, 11, 12, 14, 13, 15, 16, 11, 14, 17, 18, 12, 19, 13, 14, 5, 0, 1, 2, 6, 3, 7, 8, 15, 4, 9, 10, 16, 11, 17, 18, 20, 5, 12, 13, 19, 14, 20, 21, 21, 15

Offset: 1

Reinhard Zumkeller, Dec 21 2002

			n = 12 -> '1100': A023416(12) = 2 therefore a(12) = #{4 ->'100', 9 ->'1001', 10 ->'1010'} = 3.
n = 13 -> '1101': A023416(13) = 1 therefore a(13) = #{2 ->'10', 5 ->'101', 6 ->'110', 11 ->'1011'} = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A068076, A079071, A079072, A007088, A023416.

import Data.List (elemIndices)
a079070 n = length $ elemIndices (a023416 n) $ map a023416 [1..n-1]
-- Reinhard Zumkeller, Jun 16 2011

dcn[n_]:=Count[Range[n-1],?(DigitCount[#,2,0]==DigitCount[ n,2,0]&)]; Array[dcn,90] (* _Harvey P. Dale, Jun 11 2011 *)

a(n) = #{m : m < n and A023416(m) = A023416(n)}.

a(2^k - 1) = k - 1; a(2^k) = 0; a(2^k + 1) = 1.

A322795 Number of integers k, 0 <= k <= n, such that the Damerau-Levenshtein distance between the binary representations of n and k is strictly less than the Levenshtein distance.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 1, 2, 0, 0, 0, 1, 0, 2, 1, 3, 0, 4, 4, 4, 1, 4, 2, 4, 0, 0, 0, 1, 0, 2, 1, 3, 0, 4, 5, 5, 1, 5, 4, 7, 0, 8, 9, 9, 6, 8, 8, 8, 1, 8, 8, 8, 2, 8, 4, 8, 0, 0, 0, 1, 0, 2, 1, 4, 0, 4, 6, 6, 1, 5, 4, 9, 0, 8, 11, 11, 7, 10, 11, 11, 1, 10, 12, 13, 5, 13, 9, 14, 0, 16, 18, 17, 15, 16

Offset: 0

Pontus von Brömssen, Dec 26 2018

a(n) = 0 if and only if n appears in A099627 or n = 0.

a(n) = A079071(n) for n <= 21, but a(22) = 3 > 2 = A079071(22).

			For n = 6, the Damerau-Levenshtein distance and the Levenshtein distance between the binary representations of n and k are equal for all k <= n except k = 5. The Levenshtein distance between 101 and 110 (5 and 6 in binary) is 2, whereas the Damerau-Levenshtein distance is 1, so a(6) = 1.

Pontus von Brömssen, Table of n, a(n) for n = 0..1000
Wikipedia, Damerau-Levenshtein distance

Cf. A152487, A322285, A099627, A079071.

cp's OEIS Frontend

A079074 Sum of numbers < n having in binary representation the same number of 0's and 1's as n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A079070 Number of numbers < n having in binary representation the same number of 0's as n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A322795 Number of integers k, 0 <= k <= n, such that the Damerau-Levenshtein distance between the binary representations of n and k is strictly less than the Levenshtein distance.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs