A175054 -id:A175054 - cp's OEIS frontend

A175057 a(n) = decimal equivalent of {A175054(n) written in binary, and each run of 0's reduced in length by one digit}.

1, 3, 2, 7, 4, 5, 6, 15, 8, 9, 11, 12, 13, 14, 31, 16, 17, 19, 10, 23, 24, 25, 27, 28, 29, 30, 63, 32, 33, 35, 18, 39, 20, 21, 22, 47, 48, 49, 51, 26, 55, 56, 57, 59, 60, 61, 62, 127, 64, 65, 67, 34, 71, 36, 37, 38, 79, 40, 41, 43, 44, 45, 46, 95, 96, 97, 99, 50, 103, 52, 53, 54

Offset: 1

Leroy Quet, Dec 08 2009

This is a permutation of the positive integers. Sequence A175058 is its inverse permutation.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A175054, A175058, A175055, A175056, A175059, A175060

Map[FromDigits[#, 2] &@ Flatten@ # &, Select[Array[Split@ IntegerDigits[#, 2] &, 216], FreeQ[#, {0}] &] /. w_List /; First@ w == 0 :> Most@ w] (* Michael De Vlieger, Sep 03 2017 *)

Extended by Ray Chandler, Dec 18 2009

A022340 Even Fibbinary numbers (A003714); also 2*Fibbinary(n).

Original entry on oeis.org

0, 2, 4, 8, 10, 16, 18, 20, 32, 34, 36, 40, 42, 64, 66, 68, 72, 74, 80, 82, 84, 128, 130, 132, 136, 138, 144, 146, 148, 160, 162, 164, 168, 170, 256, 258, 260, 264, 266, 272, 274, 276, 288, 290, 292, 296, 298, 320, 322, 324, 328, 330, 336, 338, 340, 512

Offset: 0

Marc LeBrun

nonn

Positions of ones in binomial(3k+2,k+1)/(3k+2) modulo 2 (A085405). - Paul D. Hanna, Jun 29 2003
Construction: start with strings S(0)={0}, S(1)={2}; for k>=2, concatenate all prior strings excluding S(k-1) and add 2^k to each element in the resulting string to obtain S(k); this sequence is the concatenation of all such generated strings: {S(0),S(1),S(2),...}. Example: for k=5, concatenate {S(0),S(1),S(2),S(3)} = {0, 2, 4, 8,10}; add 2^5 to each element to obtain S(5)={32,34,38,40,42}. - Paul D. Hanna, Jun 29 2003
From Gus Wiseman, Apr 08 2020: (Start)
The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. This sequence lists all numbers k such that the k-th composition in standard order has no ones. For example, the sequence together with the corresponding compositions begins:
()          80: (2,5)        260: (6,3)
(2)         82: (2,3,2)      264: (5,4)
(3)         84: (2,2,3)      266: (5,2,2)
(4)        128: (8)          272: (4,5)
(2,2)      130: (6,2)        274: (4,3,2)
(5)        132: (5,3)        276: (4,2,3)
(3,2)      136: (4,4)        288: (3,6)
(2,3)      138: (4,2,2)      290: (3,4,2)
(6)        144: (3,5)        292: (3,3,3)
(4,2)      146: (3,3,2)      296: (3,2,4)
(3,3)      148: (3,2,3)      298: (3,2,2,2)
(2,4)      160: (2,6)        320: (2,7)
(2,2,2)    162: (2,4,2)      322: (2,5,2)
(7)        164: (2,3,3)      324: (2,4,3)
(5,2)      168: (2,2,4)      328: (2,3,4)
(4,3)      170: (2,2,2,2)    330: (2,3,2,2)
(3,4)      256: (9)          336: (2,2,5)
(3,2,2)    258: (7,2)        338: (2,2,3,2)
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Equals 2 * A003714.
Cf. A006013, A001045, A085405, A085407.
Compositions with no ones are counted by A212804.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Compositions without terms > 2 are A003754.
- Compositions without ones are A022340 (this sequence).
- Sum is A070939.
- Compositions with no twos are A175054.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Runs-resistance is A333628.
Cf. A066099, A124767, A228351, A318928, A333218.

a022340 = (* 2) . a003714 -- Reinhard Zumkeller, Feb 03 2015

f[n_Integer] := Block[{k = Ceiling[ Log[ GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k-- ]; FromDigits[fr, 2]]; Select[f /@ Range[0, 95], EvenQ[ # ] &] (* Robert G. Wilson v, Sep 18 2004 *)
Select[Range[2, 512, 2], BitAnd[#, 2#] == 0 &] (* Alonso del Arte, Jun 18 2012 *)

from itertools import count, islice
def A022340_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:not n&(n>>1),count(max(0,startvalue+(startvalue&1)),2))
A022340_list = list(islice(A022340_gen(),30)) # Chai Wah Wu, Sep 07 2022

def A022340(n):
    tlist, s = [1,2], 0
    while tlist[-1]+tlist[-2] <= n: tlist.append(tlist[-1]+tlist[-2])
    for d in tlist[::-1]:
        if d <= n:
            s += 1
            n -= d
        s <<= 1
    return s # Chai Wah Wu, Apr 24 2025

For n>0, a(F(n))=2^n, a(F(n)-1)=A001045(n+2)-1, where F(n) is the n-th Fibonacci number with F(0)=F(1)=1.

a(n) + a(n)/2 = a(n) XOR a(n)/2, see A106409. - Reinhard Zumkeller, May 02 2005

Edited by Ralf Stephan, Sep 01 2004

A377167 Nonnegative integers containing isolated zeros in their binary representation.

Original entry on oeis.org

0, 2, 5, 6, 10, 11, 13, 14, 18, 20, 21, 22, 23, 26, 27, 29, 30, 34, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 50, 52, 53, 54, 55, 58, 59, 61, 62, 66, 69, 70, 74, 75, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 98, 101, 102, 104, 105, 106

Offset: 1

Paolo Xausa, Oct 18 2024

A 0 is isolated if it's not adjacent to another 0.

			18 is a term because 18 = 10010_2 contains one isolated 0.
85 is a term because all zeros in 85 = 1010101_2 are isolated.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Complement of A175054.

Cf. A004748, A007088, A377169.

Select[Range[0, 150], MemberQ[Split[IntegerDigits[#, 2]], {0}] &]

{ k : k=0 or 2k+1 in {A004748} }.

cp's OEIS Frontend

A175057 a(n) = decimal equivalent of {A175054(n) written in binary, and each run of 0's reduced in length by one digit}.

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A022340 Even Fibbinary numbers (A003714); also 2*Fibbinary(n).

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A377167 Nonnegative integers containing isolated zeros in their binary representation.

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Mathematica

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