A227351 Permutation of nonnegative integers: map each number by lengths of runs of zeros in its Zeckendorf expansion shifted once left to the number which has the same lengths of runs (in the same order, but alternatively of runs of 0's and 1's) in its binary representation.

0, 1, 3, 7, 2, 15, 6, 4, 31, 14, 12, 8, 5, 63, 30, 28, 24, 13, 16, 9, 11, 127, 62, 60, 56, 29, 48, 25, 27, 32, 17, 19, 23, 10, 255, 126, 124, 120, 61, 112, 57, 59, 96, 49, 51, 55, 26, 64, 33, 35, 39, 18, 47, 22, 20, 511, 254, 252, 248, 125, 240, 121, 123, 224

Offset

0,3

Comments

This permutation is based on the fact that by appending one extra zero to the right of Fibonacci number representation of n (aka "Zeckendorf expansion") and then counting the lengths of blocks of consecutive (nonleading) zeros we get bijective correspondence with compositions, and thus also with the binary representation of a unique n. See the chart below:

n A014417(n) A014417(A022342(n+1)) Runs of Binary number In dec.

[shifted once left] zeros with same runs = a(n)

0: ......0 ......0 [] .....0 0

1: ......1 .....10 [1] .....1 1

2: .....10 ....100 [2] ....11 3

3: ....100 ...1000 [3] ...111 7

4: ....101 ...1010 [1,1] ....10 2

5: ...1000 ..10000 [4] ..1111 15

6: ...1001 ..10010 [2,1] ...110 6

7: ...1010 ..10100 [1,2] ...100 4

8: ..10000 .100000 [5] .11111 31

9: ..10001 .100010 [3,1] ..1110 14

10: ..10010 .100100 [2,2] ..1100 12

11: ..10100 .101000 [1,3] ..1000 8

12: ..10101 .101010 [1,1,1] ...101 5

13: .100000 1000000 [6] 111111 63

Are there any other fixed points after 0, 1, 6, 803, 407483 ?

Links

Antti Karttunen, Table of n, a(n) for n = 0..10946

OEIS Wiki, Run-length encoding

Index entries for sequences that are permutations of the natural numbers

Formula

a(n) = A006068(A048679(n)) = A006068(A106151(2*A003714(n))).

This permutation effects following correspondences:

a(A000045(n)) = A000225(n-1).

a(A027941(n)) = A000975(n).

For n >=3, a(A000204(n)) = A000079(n-2).

Prog

(Scheme): (define (A227351 n) (A006068 (A048679 n)))
(define (A227351v2 n) (A006068 (A106151 (* 2 (A003714 n))))) ;; Alternative definition.

Crossrefs

Inverse permutation: A227352. Cf. also A003714, A014417, A006068, A048679.

Could be further composed with A075157 or A075159, also A129594.

Sequence in context: A260433 A243343 A255565 * A246377 A260421 A237427

Adjacent sequences: A227348 A227349 A227350 * A227352 A227353 A227354

Keyword

nonn,base

Author

Antti Karttunen, Jul 08 2013

Status

approved