A106151 -id:A106151 - cp's OEIS frontend

A048679 Compressed fibbinary numbers (A003714), with rewrite 0->0, 01->1 applied to their binary expansion.

0, 1, 2, 4, 3, 8, 5, 6, 16, 9, 10, 12, 7, 32, 17, 18, 20, 11, 24, 13, 14, 64, 33, 34, 36, 19, 40, 21, 22, 48, 25, 26, 28, 15, 128, 65, 66, 68, 35, 72, 37, 38, 80, 41, 42, 44, 23, 96, 49, 50, 52, 27, 56, 29, 30, 256, 129, 130, 132, 67, 136, 69, 70, 144, 73, 74, 76, 39, 160, 81

Offset: 0

Antti Karttunen

Permutation of the nonnegative integers (A001477); inverse permutation of A048680 i.e. A048679[ A048680[ n ] ] = n for all n.

Alois P. Heinz, Table of n, a(n) for n = 0..17710 (terms n = 10001..10945 from Antti Karttunen)
Index entries for sequences that are permutations of the natural numbers

Cf. A000045, A003714, A005203, A048678, A048680, A072650, A087808, A106151, A200714, A227351, A232559, A277006, A304100, A304101.

a(n) = rewrite_0to0_x1to1(fibbinary(j)) (where fibbinary(j) = A003714[ n ])
rewrite_0to0_x1to1 := proc(n) option remember; if(0 = n) then RETURN(n); else RETURN((2 * rewrite_0to0_x1to1(floor(n/(2^(1+(n mod 2)))))) + (n mod 2)); fi; end;
fastfib := n -> round((((sqrt(5)+1)/2)^n)/sqrt(5)); fibinv_appr := n -> floor(log[ (sqrt(5)+1)/2 ](sqrt(5)*n)); fibinv := n -> (fibinv_appr(n) + floor(n/fastfib(1+fibinv_appr(n)))); fibbinary := proc(n) option remember; if(n <= 2) then RETURN(n); else RETURN((2^(fibinv(n)-2))+fibbinary_seq(n-fastfib(fibinv(n)))); fi; end;
# second Maple program:
b:= proc(n) is(n=0) end:
a:= proc(n) option remember; local h; h:= iquo(a(n-1), 2)+1;
      while b(h) do h:= h*2 od; b(h):=true; h
    end: a(0):=0:
seq(a(n), n=0..100);  # Alois P. Heinz, Sep 22 2014

b[n_] := n==0; a[n_] := a[n] = Module[{h}, h = Quotient[a[n-1], 2] + 1; While[b[h], h = h*2]; b[h] = True; h]; a[0]=0; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 27 2016, after Alois P. Heinz *)

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A007814(n) = valuation(n,2);
A000265(n) = (n/2^valuation(n, 2));
A106151(n) = if(n<=1,n,if(n%2,1+(2*A106151((n-1)/2)),(2^(A007814(n)-1))*A106151(A000265(n))));
A048679(n) = if(!n,n,A106151(2*A003714(n))); \\ Antti Karttunen, May 13 2018, after Reinhard Zumkeller's May 09 2005 formula.

from itertools import count, islice
def A048679_gen(): # generator of terms
    return map(lambda n: int(bin(n)[2:].replace('01','1'),2),filter(lambda n:not (n<<1)&n,count(0)))
A048679_list = list(islice(A048679_gen(),20)) # Chai Wah Wu, Mar 18 2024

def A048679(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
        else:
            s <<= 1
    return s # Chai Wah Wu, Apr 24 2025

a(n) = A106151(2*A003714(n)) for n > 0. - Reinhard Zumkeller, May 09 2005
a(n+1) = min{([a(n)/2]+1)*2^k} such that it is not yet in the sequence. - Gerard Orriols, Jun 07 2014
a(n) = A072650(A003714(n)) = A003188(A227351(n)). - Antti Karttunen, May 13 2018

A304101 Restricted growth sequence transform of A278222(A048679(n)).

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 4, 3, 2, 4, 4, 3, 5, 2, 4, 4, 4, 6, 3, 6, 5, 2, 4, 4, 4, 6, 4, 7, 6, 3, 6, 6, 5, 8, 2, 4, 4, 4, 6, 4, 7, 6, 4, 7, 7, 6, 9, 3, 6, 6, 6, 10, 5, 9, 8, 2, 4, 4, 4, 6, 4, 7, 6, 4, 7, 7, 6, 9, 4, 7, 7, 7, 11, 6, 11, 9, 3, 6, 6, 6, 10, 6, 11, 10, 5, 9, 9, 8, 12, 2, 4, 4, 4, 6, 4, 7, 6, 4, 7, 7, 6, 9, 4, 7, 7, 7

Offset: 0

Antti Karttunen, May 13 2018

Positions of 2's is given by the positive Fibonacci numbers: 1, 2, 3, 5, 8, 13, 21, ..., that is, A000045(n) from n >= 2 onward.
Positions of 3's is given by Lucas numbers larger than 3: 4, 7, 11, 18, ..., that is, A000032(n) from n >= 3 onward.
Sequence allots a distinct value for each distinct multiset formed from the lengths of 1-runs in the binary representation of A048679(n). Compare to the scatter plot of A286622.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A000032, A000045, A003603, A007895, A035513, A048679, A278222, A304100, A304102, A304103, A304104, A304105.

Cf. also A286622 (compare the scatter-plots).

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A106151(n) = if(n<=1, n, if(n%2, 1+(2*A106151((n-1)/2)), A106151(n>>valuation(n, 2))<<(valuation(n, 2)-1)));
A048679(n) = if(!n,n,A106151(2*A003714(n)));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
v304101 = rgs_transform(vector(1+up_to, n, A278222(A048679(n-1))));
A304101(n) = v304101[1+n];

A175047 Write n in binary, then increase each run of 0's by one 0. a(n) is the decimal equivalent of the result.

Original entry on oeis.org

1, 4, 3, 8, 9, 12, 7, 16, 17, 36, 19, 24, 25, 28, 15, 32, 33, 68, 35, 72, 73, 76, 39, 48, 49, 100, 51, 56, 57, 60, 31, 64, 65, 132, 67, 136, 137, 140, 71, 144, 145, 292, 147, 152, 153, 156, 79, 96, 97, 196, 99, 200, 201, 204, 103, 112, 113, 228, 115, 120, 121, 124, 63, 128

Offset: 1

Leroy Quet, Dec 02 2009

From Reinhard Zumkeller, Dec 12 2009: (Start)
A070939(a(n)) = A070939(n) + A033264(n);
A171598 and A171599 give record values and where they occur. (End)

			12 in binary is 1100. Increase each run of 0 by one digit to get 11000, which is 24 in decimal. So a(12) = 24.

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A175046, A175048.
Cf. A084471. - Robert G. Wilson v, Dec 11 2009
Cf. A030308, A106151, A007088.

import Data.List (group)
a175047 = foldr (\b v -> 2 * v + b) 0 . concatMap
   (\bs@(b:_) -> if b == 0 then 0 : bs else bs) . group . a030308_row
-- Reinhard Zumkeller, Jul 05 2013

f[n_] := Block[{s = Split@ IntegerDigits[n, 2]}, FromDigits[ Flatten@ Insert[s, {0}, Table[{2 i}, {i, Floor[ Length@s/2]} ]], 2]]; Array[ f, 64] (* Robert G. Wilson v, Dec 11 2009 *)

from re import split
def A175047(n):
  return int(''.join(d+'0' if '0' in d else d for d in split('(0+)|(1+)',bin(n)[2:]) if d != '' and d != None),2) # Chai Wah Wu, Nov 21 2018

a(n) = if n<2 then n else 2*(1 + 0^((n+2) mod 4))*a([n/2]) + n mod 2. - Reinhard Zumkeller, Jan 20 2010

a(2^n) = 2^(n+1). - Chai Wah Wu, Nov 21 2018

Extended by Ray Chandler, Dec 18 2009

a(11) onwards from Robert G. Wilson v and Reinhard Zumkeller, Dec 11 2009

A348710 In the binary expansion of n, decrease the length of each run of 1-bits by one.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 3, 0, 0, 0, 1, 4, 2, 6, 7, 0, 0, 0, 1, 0, 0, 2, 3, 8, 4, 4, 5, 12, 6, 14, 15, 0, 0, 0, 1, 0, 0, 2, 3, 0, 0, 0, 1, 4, 2, 6, 7, 16, 8, 8, 9, 8, 4, 10, 11, 24, 12, 12, 13, 28, 14, 30, 31, 0, 0, 0, 1, 0, 0, 2, 3, 0, 0, 0, 1, 4, 2, 6, 7, 0, 0, 0

Offset: 0

Kevin Ryde, Oct 30 2021

Equivalently, change bits 01 -> 0, including a 0 reckoned above the most significant 1-bit of n so change there.
A single 1-bit run decreases to nothing.  The Fibbinary numbers (A003714) are those n with only single 1-bits so that a(n) = 0 iff n is in A003714.
a(n) = 1 iff n is in A213540 since those values end with bits 011 (which become 01) and otherwise have only single 1-bits, as do the Fibbinary numbers.
Decreasing each run is the inverse of the increase A175048 so that a(A175048(k)) = k.  This n = A175048(k) is the smallest n with a(n) = k and then other occurrences of k are by inserting single 1-bits into this n, including anywhere above the most significant bit.

			n    = 14551 = binary 111 000 11 0 1 0 111
a(n) =   787 = binary  11 000  1 0   0  11

Cf. A007088 (binary), A175048 (increase 1-bits), A090077 (decrease to single 1-bits).
Cf. A003714 (indices of 0's), A213540 (indices of 1's).
Cf. A106151 (decrease 0-bits), A318921 (decrease each run).

Table[FromDigits[Flatten[Split@IntegerDigits[n,2]/. {1,a___}:>{a}],2],{n,0,82}] (* Giorgos Kalogeropoulos, Nov 01 2021 *)

a(n) = my(v=binary(n),t=0); for(i=2,#v, if(v[i-1]||!v[i], v[t++]=v[i])); fromdigits(v[1..t],2);

def a(n): return int(bin(n).replace("b", "").replace("01", "0"), 2)
print([a(n) for n in range(83)]) # Michael S. Branicky, Oct 31 2021

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).

A304100 a(n) = A003602(A048679(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 1, 5, 3, 2, 4, 1, 9, 5, 3, 6, 2, 7, 4, 1, 17, 9, 5, 10, 3, 11, 6, 2, 13, 7, 4, 8, 1, 33, 17, 9, 18, 5, 19, 10, 3, 21, 11, 6, 12, 2, 25, 13, 7, 14, 4, 15, 8, 1, 65, 33, 17, 34, 9, 35, 18, 5, 37, 19, 10, 20, 3, 41, 21, 11, 22, 6, 23, 12, 2, 49, 25, 13, 26, 7, 27, 14, 4, 29, 15, 8, 16, 1, 129, 65, 33, 66, 17, 67, 34, 9, 69, 35, 18, 36, 5, 73

Offset: 1

Antti Karttunen, May 13 2018

nonn

Positions of ones is given by the positive Fibonacci numbers: 1, 2, 3, 5, 8, 13, 21, ..., that is, A000045(n) from n >= 2 onward.
Positions of 2's is given by Lucas numbers larger than 3: 4, 7, 11, 18, ..., that is, A000032(n) from n >= 3 onward.
The restricted growth sequence transform of this sequence (almost certainly) is A003603.

Antti Karttunen, Table of n, a(n) for n = 1..17710

Cf. A000032, A000045, A003602, A003603, A035513, A048679, A304101.

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A007814(n) = valuation(n,2);
A000265(n) = (n/2^valuation(n, 2));
A106151(n) = if(n<=1,n,if(n%2,1+(2*A106151((n-1)/2)),(2^(A007814(n)-1))*A106151(A000265(n))));
A048679(n) = if(!n,n,A106151(2*A003714(n)));
A003602(n) = (1+A000265(n))/2;
A304100(n) = A003602(A048679(n));

a(n) = A003602(A048679(n)).

For all i, j: a(i) = a(j) => A304101(i) = A304101(j).

A374201 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A048679(A328845(i))) = A278222(A048679(A328845(j))), for all i, j >= 1, where A328845 is a Fibonacci-based variant of the arithmetic derivative.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 3, 2, 4, 4, 5, 2, 4, 2, 6, 7, 4, 2, 6, 2, 8, 9, 10, 2, 4, 7, 8, 6, 11, 2, 8, 2, 11, 8, 12, 7, 4, 2, 12, 13, 7, 2, 11, 2, 8, 8, 13, 2, 7, 10, 7, 8, 14, 2, 15, 7, 13, 12, 10, 2, 8, 2, 12, 7, 10, 13, 5, 2, 16, 17, 5, 2, 7, 2, 13, 7, 15, 16, 18, 2, 7, 18, 12, 2, 8, 19, 20, 13, 7, 2, 8, 18, 16, 12, 10, 21, 13, 2, 8, 9, 16

Offset: 0

Antti Karttunen, Jul 02 2024

nonn

Restricted growth sequence transform of A278222(A048679(A328845(n))), or equally, of A304101(A328845(n)).
Related to the Zeckendorf-representation (A014417) of A328845(n).
For all i, j >= 0: a(i) = a(j) => A328847(i) = A328847(j).

Antti Karttunen, Table of n, a(n) for n = 0..75025

Cf. A000045, A014417, A048679, A278222, A304101, A328845, A328847.

Cf. also A304105, A373982.

up_to = 75025;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A106151(n) = { my(s=0, i=0); while(n, if(2!=(n%4), s += (n%2)<>= 1); (s); };
A048679(n) = if(!n,n,A106151(2*A003714(n)));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
A278222(n) = A046523(A005940(1+n));
A328845(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])/f[i, 1]));
v374201 = rgs_transform(vector(1+up_to, n, A278222(A048679(A328845(n-1)))));
A374201(n) = v374201[1+n];

cp's OEIS Frontend

A048679 Compressed fibbinary numbers (A003714), with rewrite 0->0, 01->1 applied to their binary expansion.

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A304101 Restricted growth sequence transform of A278222(A048679(n)).

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A175047 Write n in binary, then increase each run of 0's by one 0. a(n) is the decimal equivalent of the result.

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A348710 In the binary expansion of n, decrease the length of each run of 1-bits by one.

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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.

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A304100 a(n) = A003602(A048679(n)).

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A374201 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A048679(A328845(i))) = A278222(A048679(A328845(j))), for all i, j >= 1, where A328845 is a Fibonacci-based variant of the arithmetic derivative.

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