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A022340
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Even Fibbinary numbers (A003714); also 2*Fibbinary(n).
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22
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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
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OFFSET
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0,2
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COMMENTS
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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
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)
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LINKS
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FORMULA
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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.
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
(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))
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CROSSREFS
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Compositions with no ones are counted by A212804.
All of the following pertain to compositions in standard order (A066099):
- Compositions without terms > 2 are A003754.
- Compositions without ones are A022340 (this sequence).
- Compositions with no twos are A175054.
- Constant compositions are A272919.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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