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

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

Author

Marc LeBrun

Keywords

• nonn

Comments

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:

0: () 80: (2,5) 260: (6,3)

2: (2) 82: (2,3,2) 264: (5,4)

4: (3) 84: (2,2,3) 266: (5,2,2)

8: (4) 128: (8) 272: (4,5)

10: (2,2) 130: (6,2) 274: (4,3,2)

16: (5) 132: (5,3) 276: (4,2,3)

18: (3,2) 136: (4,4) 288: (3,6)

20: (2,3) 138: (4,2,2) 290: (3,4,2)

32: (6) 144: (3,5) 292: (3,3,3)

34: (4,2) 146: (3,3,2) 296: (3,2,4)

36: (3,3) 148: (3,2,3) 298: (3,2,2,2)

40: (2,4) 160: (2,6) 320: (2,7)

42: (2,2,2) 162: (2,4,2) 322: (2,5,2)

64: (7) 164: (2,3,3) 324: (2,4,3)

66: (5,2) 168: (2,2,4) 328: (2,3,4)

68: (4,3) 170: (2,2,2,2) 330: (2,3,2,2)

72: (3,4) 256: (9) 336: (2,2,5)

74: (3,2,2) 258: (7,2) 338: (2,2,3,2)

(End)

Links

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

Crossrefs

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.

Programs

Haskell

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

Mathematica

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 *)

Python

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

Python

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

Formula

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

Extensions

Edited by Ralf Stephan, Sep 01 2004