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

0,3

Comments

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

Links

Antti Karttunen, Table of n, a(n) for n = 0..10945 (terms 0..10000 from Alois P. Heinz)

Index entries for sequences that are permutations of the natural numbers

Formula

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

Maple

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

Mathematica

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 *)

Prog

(PARI)
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.
(Python)
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

Crossrefs

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

Sequence in context: A225589 A245603 A371591 * A342794 A266412 A246365

Adjacent sequences: A048676 A048677 A048678 * A048680 A048681 A048682

Keyword

nonn,base

Author

Antti Karttunen

Status

approved