A030302 Write n in base 2 and juxtapose; irregular table in which row n lists the binary expansion of n.

1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• base
• cons
• easy
• tabf

Comments

The binary Champernowne constant: it is normal in base 2. - Jason Kimberley, Dec 07 2012

A word that is recurrent, but neither morphic nor uniformly recurrent. - N. J. A. Sloane, Jul 14 2018

See A030303 for the indices of 1's (so this is the characteristic function of A030303), with first differences (i.e., run lengths of 0's, increased by 1, with two consecutive 1's delimiting a run of zero 0's) given by A066099. - M. F. Hasler, Oct 12 2020

References

• Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

Links

• R. J. Mathar, Table of n, a(n) for n = 1..3998
• Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit and Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017.

Crossrefs

Essentially the same as A007088 and A030190. Cf. A030303, A007088.

Tables in which the n-th row lists the base b digits of n: A030190 and this sequence (b=2), A003137 and A054635 (b=3), A030373 (b=4), A031219 (b=5), A030548 (b=6), A030998 (b=7), A031035 and A054634 (b=8), A031076 (b=9), A007376 and A033307 (b=10). [Jason Kimberley, Dec 06 2012]

Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: A316829, 37: A010060.

Programs

Magma

&cat[Reverse(IntegerToSequence(n,2)): n in [1..31]]; // Jason Kimberley, Mar 02 2012

Maple

A030302 := proc(n) local i,t1,t2; t1:=convert(n,base,2); t2:=nops(t1); [seq(t1[t2+1-i],i=1..t2)]; end; # N. J. A. Sloane, Apr 08 2021

Mathematica

i[n_] := Ceiling[FullSimplify[ProductLog[Log[2]/2 (n - 1)]/Log[2] + 1]]; a[n_] := Mod[Floor[2^(Mod[n + 2^i[n] - 2, i[n]] - i[n] + 1) Ceiling[(n + 2^i[n] - 1)/i[n] - 1]], 2]; (* David W. Cantrell (DWCantrell(AT)sigmaxi.net), Feb 19 2007 *)
Join @@ Table[ IntegerDigits[i, 2], {i, 1, 40}] (* Olivier Gérard, Mar 28 2011 *)
Flatten@ IntegerDigits[ Range@ 25, 2] (* or *)
almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; Array[ almostNatural[#, 2] &, 105] (* Robert G. Wilson v, Jun 29 2014 *)

Python

from itertools import count, islice
def A030302_gen(): # generator of terms
    return (int(d) for n in count(1) for d in bin(n)[2:])
A030302_list = list(islice(A030302_gen(),30)) # Chai Wah Wu, Feb 18 2022

Formula

a(n) = (floor(2^(((n + 2^i - 2) mod i) - i + 1) * ceiling((n + 2^i - 1)/i - 1))) mod 2 where i = ceiling( W(log(2)/2 (n - 1))/log(2) + 1 ) and W denotes the principal branch of the Lambert W function. See also Mathematica code. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Feb 19 2007