A048679 -id:A048679 - cp's OEIS frontend

A072650 Starting from the right (the least significant end) rewrite 0 to 0 and x1 to 1 in the binary expansion of n.

0, 1, 2, 1, 4, 3, 2, 3, 8, 5, 6, 5, 4, 3, 6, 3, 16, 9, 10, 9, 12, 7, 10, 7, 8, 5, 6, 5, 12, 7, 6, 7, 32, 17, 18, 17, 20, 11, 18, 11, 24, 13, 14, 13, 20, 11, 14, 11, 16, 9, 10, 9, 12, 7, 10, 7, 24, 13, 14, 13, 12, 7, 14, 7, 64, 33, 34, 33, 36, 19, 34, 19, 40, 21, 22, 21, 36, 19, 22, 19

Offset: 0

Antti Karttunen, Jun 02 2002

nonn

			I.e. 23 is 10111 in binary, which after rewriting is 111, thus a(23) = 7, while 38 is 100110 in binary, which after the rewriting is 10010, i.e. a(38) = 18.

Cf. A003714, A048679.

A048679(n) = a(A003714(n)).

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.

Original entry on oeis.org

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

Antti Karttunen, Jul 08 2013

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
  ......1              .....10    [1]        .....1           1
  .....10              ....100    [2]        ....11           3
  ....100              ...1000    [3]        ...111           7
  ....101              ...1010    [1,1]      ....10           2
  ...1000              ..10000    [4]        ..1111          15
  ...1001              ..10010    [2,1]      ...110           6
  ...1010              ..10100    [1,2]      ...100           4
  ..10000              .100000    [5]        .11111          31
  ..10001              .100010    [3,1]      ..1110          14
  ..10010              .100100    [2,2]      ..1100          12
  ..10100              .101000    [1,3]      ..1000           8
  ..10101              .101010    [1,1,1]    ...101           5
  .100000              1000000    [6]        111111          63
Are there any other fixed points after 0, 1, 6, 803, 407483 ?

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

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

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

(define (A227351 n) (A006068 (A048679 n)))

(define (A227351v2 n) (A006068 (A106151 (* 2 (A003714 n)))))

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).

A056017 Permutation of nonnegative integers formed by ranking fibbinary numbers (A003714) as if they were representatives of the circular binary sequences with forbidden -11- subsequence.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 11, 9, 10, 12, 13, 18, 14, 15, 19, 16, 20, 17, 21, 29, 22, 23, 30, 24, 31, 25, 26, 32, 27, 28, 33, 34, 47, 35, 36, 48, 37, 49, 38, 39, 50, 40, 41, 51, 42, 52, 43, 44, 53, 45, 54, 46, 55, 76, 56, 57, 77, 58, 78, 59, 60, 79, 61, 62, 80, 63, 81, 64, 65

Offset: 0

Antti Karttunen, Jun 08 2000

nonn

Function CircBinSeqNo11Rank gives the position of any 11-free binary sequence in this sequence, where each block consists of Lucas(n-2) sequences of length n: (either the leftmost or the rightmost digit is 1, but not both).

In this permutation the Fibonacci numbers themselves (A000045) are fixed.

			0; 01,10; 100; 0101,1000,1010; 01001,10000,10010,10100; 010001,010101,100000,100010,100100,101000,101010; etc.

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A056018. For fibbinary function see A048679, interpret_as_zeckendorf_expansion given in A048680.

CircBinSeqNo11Rank := n -> fibonacci(floor_log_2(n)+1-((-1)^n)) + interpret_as_zeckendorf_expansion(floor(n/(3-((-1)^n))));

[seq(CycBinSeqNo11Rank(fibbinary(j)), j=0..233)];

a[0] = 0, a[n] = CircBinSeqNo11Rank(fibbinary(n)) for n >= 1.

A072794 Inverse permutation to A072793.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 10, 8, 9, 12, 11, 14, 15, 16, 21, 13, 23, 17, 18, 36, 19, 20, 25, 22, 27, 28, 29, 40, 30, 31, 38, 24, 44, 45, 46, 55, 26, 57, 32, 33, 78, 34, 35, 42, 37, 82, 49, 50, 59, 39, 61, 47, 48, 136, 53, 54, 63, 56, 65, 66, 67, 86, 68, 69, 80, 41, 90, 91, 92

Offset: 0

Antti Karttunen, Jun 12 2002

nonn

Index entries for sequences that are permutations of the natural numbers

Composition of A054239 & A048679.

A379658 The balanced ternary expansion of a(n) is obtained by removing a digit 0 before each nonzero digit in the nonadjacent form of n.

Original entry on oeis.org

0, 1, 3, 2, 9, 4, 6, 8, 27, 10, 12, 5, 18, 7, 24, 26, 81, 28, 30, 11, 36, 13, 15, 17, 54, 19, 21, 23, 72, 25, 78, 80, 243, 82, 84, 29, 90, 31, 33, 35, 108, 37, 39, 14, 45, 16, 51, 53, 162, 55, 57, 20, 63, 22, 69, 71, 216, 73, 75, 77, 234, 79, 240, 242, 729

Offset: 0

Rémy Sigrist, Dec 29 2024

A permutation of the nonnegative integers with inverse A379657.

			The first terms are:
  n   a(n)  naf(n)  bter(a(n))
  --  ----  ------  ----------
   0     0       0           0
   1     1       1           1
   2     3      10          10
   3     2     10T          1T
   4     9     100         100
   5     4     101          11
   6     6    10T0         1T0
   7     8    100T         10T
   8    27    1000        1000
   9    10    1001         101
  10    12    1010         110
  11     5   10T0T         1TT
  12    18   10T00        1T00

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Joerg Arndt, Matters Computational (The Fxtbook), pages 61-62.
Wikipedia, Balanced ternary
Wikipedia, Non-adjacent form
Index entries for sequences that are permutations of the natural numbers

See A048679 for a similar sequence.

Cf. A065363, A334913, A379657 (inverse).

a(n) = { my (v = 0, t = 1, d); while (n, if (n%2, n -= d = 2 - (n%4); v += d*t; t /= 3;); n \= 2; t *= 3;); return (v); }

A065363(a(n)) = A334913(n).

A358733 Permutation of the nonnegative integers such that A358654(p(n) - 1) = A200714(n) for n > 0 where p(n) is described in Comments.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 11, 10, 12, 13, 17, 14, 18, 15, 16, 19, 20, 21, 22, 27, 23, 28, 29, 24, 25, 26, 30, 31, 32, 33, 34, 43, 35, 44, 36, 37, 45, 46, 47, 38, 39, 40, 41, 42, 49, 48, 50, 51, 52, 53, 54, 55, 56, 69, 57, 70, 71, 58, 59, 60, 72, 73, 74, 75

Offset: 0

Mikhail Kurkov, Mar 13 2023 [verification needed]

Here p(n) = n + a(d(n)) - d(n) for n > 0 where d(n) = c(b(n)), b(n) = f(g(n) + 2) - n - 1 for n > 0 with b(0) = 0, c(n) = f(g(n) + 3) - n - 1 for n > 0 with c(0) = 0, f(n) = A000045(n) and where g(n) = A072649(n). To compute p(n) we need to know a(d(n)) and to compute a(n) we need to know p(e(n)) where e(n) = n - f(g(n) + 1) for n > 0 with e(0) = 0 in the sense that we can rewrite a(n) = n + [e(n) > 0]*(a(h(n)) - h(n) ...) (here h(n) = d(e(n))) as a(n) = n - e(n) + [e(n) > 0]*(p(e(n)) ...).

Index entries for sequences that are permutations of the nonnegative integers

Cf. A000045, A048679, A072649, A200648, A200714, A348366, A358654.

g(n)=local(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2) \\ from A072649
d(n) = { while(n>0, my(A=g(n), B=fibonacci(A)); n-=B; if(B>n, break)); n; }
a(n) = if(n>0, my(A=g(n), B=fibonacci(A+1), C=n-B, D=d(C), E=g(C-1)); n + if(C>0, a(D) - D - fibonacci(E) + if(E%2==A%2, fibonacci(A-2))))

a(n) = n + [e(n) > 0]*(a(h(n)) - h(n) - f(s(n)) + [s(n) mod 2 = g(n) mod 2]*f(g(n) - 2)) for n > 0 with a(0) = 0 where s(n) = g(e(n) - 1) (here we also consider that g(0) = 0), h(n) = d(e(n)), e(n) = n - f(g(n) + 1) for n > 0 with e(0) = 0, d(n) = c(b(n)), b(n) = f(g(n) + 2) - n - 1 for n > 0 with b(0) = 0, c(n) = f(g(n) + 3) - n - 1 for n > 0 with c(0) = 0, f(n) = A000045(n) and where g(n) = A072649(n).

A371163 Numbers that remain unchanged when converted to their compressed fibbinary numbers.

Original entry on oeis.org

0, 1, 2, 9, 10, 115544, 13568075, 13568077, 13568078, 13568083, 13568085, 13568086

Offset: 1

Douglas Boffey, Mar 13 2024

			9 is a term since 9 = 8 + 1 = F(6) + F(2), where F(i) is the i-th Fibonacci number, is Fibbinary A003714(9) = 10001_2, but then all '01's are compressed to '1', leaving A048679(9) = 1001_2, which is 9 itself again.

Fixed points of A048679 and A048680.

Cf. A000045, A003714.

from itertools import count, islice
def A371163_gen(): # generator of terms
    c = 0
    for n in count(0):
        if not (n<<1)&n:
            if  int(bin(n)[2:].replace('01','1'),2) == c:
                yield c
            c += 1
A371163_list = list(islice(A371163_gen(),6)) # Chai Wah Wu, Mar 18 2024

cp's OEIS Frontend

A072650 Starting from the right (the least significant end) rewrite 0 to 0 and x1 to 1 in the binary expansion of n.

Views

Author

Keywords

Examples

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Scheme

Formula

A056017 Permutation of nonnegative integers formed by ranking fibbinary numbers (A003714) as if they were representatives of the circular binary sequences with forbidden -11- subsequence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A072794 Inverse permutation to A072793.

Views

Author

Keywords

Links

Crossrefs

A379658 The balanced ternary expansion of a(n) is obtained by removing a digit 0 before each nonzero digit in the nonadjacent form of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A358733 Permutation of the nonnegative integers such that A358654(p(n) - 1) = A200714(n) for n > 0 where p(n) is described in Comments.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A371163 Numbers that remain unchanged when converted to their compressed fibbinary numbers.

Views

Author

Keywords

Examples

Crossrefs

Programs

Python