A001468 There are a(n) 2's between successive 1's.

1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

The Fibonacci word on the alphabet {2,1}, with an extra 1 in front. - Michel Dekking, Nov 26 2018

Start with 1, apply 1->12, 2->122, take limit. - Philippe Deléham, Sep 23 2005

Also number of occurrences of n in Hofstadter G-sequence (A005206) and in A019446. - Reinhard Zumkeller, Feb 02 2012, Aug 07 2011

A block-fractal sequence: every block occurs infinitely many times. Also a reverse block-fractal sequence. See A280511. - Clark Kimberling, Jan 06 2017

References

• D. Gault and M. Clint, "Curiouser and curiouser" said Alice. Further reflections on an interesting recursive function, Internat. J. Computer Math., 26 (1988), 35-43. See Table 2.
• D. R. Hofstadter, personal communication, Jul 15 1977.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• M. Bunder and K. Tognetti, On the self matching properties of [j tau], Discrete Math., 241 (2001), 139-151.
• D. Gault and M. Clint, "Curiouser and curiouser" said Alice. Further reflections on an interesting recursive function, Internat. J. Computer Math., 26 (1988), 35-43.
• D. Gault and M. Clint, "Curiouser and curiouser said Alice. Further reflections on an interesting recursive function, Intern. J. Computer. Math., 26 (1988), 35-43. (Annotated scanned copy)
• D. R. Hofstadter, Eta-Lore [Cached copy, with permission]
• D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission]
• D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991 [See DRH latter, p. 2, Eq. (2), the sequence marked A006336, which is now A001468].
• J. V. Pennington and T. F. Mulcrone, Problem E1226, Amer. Math. Monthly, 64 (1957), 197-198.
• Leon Recht, Martin Rosenbaum and E. P. Starke, Problem 4247, Amer. Math. Monthly, 55 (1948), 588-592.
• N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
• N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Crossrefs

Same as A014675 if initial 1 is deleted. Cf. A003849, A000201, A280511.

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

Programs

Haskell

import Data.List (group)
a001468 n = a001468_list !! n
a001468_list = map length $ group a005206_list
-- Reinhard Zumkeller, Aug 07 2011

Maple

Digits := 100: t := evalf( (1+sqrt(5))/2); A001468 := n-> floor((n+1)*t)-floor(n*t);

Mathematica

Table[Floor[GoldenRatio*(n + 1)] - Floor[GoldenRatio*n], {n, 0, 80}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006 *)
Nest[ Flatten[# /. {1 -> {1, 2}, 2 -> {1, 2, 2}}] &, {1}, 6] (* Robert G. Wilson v, May 20 2014 and corrected Apr 24 2017 following Clark Kimberling's email of Mar 22 2017 *)
SubstitutionSystem[{1->{1,2},2->{1,2,2}},{1},{6}][[1]] (* Harvey P. Dale, Jan 31 2022 *)

PARI

a=[1];for(i=1,30,a=concat([a,vector(a[i],j,2),1]));a \\ Or compute as A001468(n)=A201(n+1)-A201(n) with A201(n)=(n+sqrtint(5*n^2))\2, working for n>=0 although A000201 is defined for n>=1. - M. F. Hasler, Oct 13 2017

Python

def A001468(length):
    a = [1]
    for i in range(length):
        for _ in range(a[i]):
            a.append(2)
        a.append(1)
        if len(a)>=length:
            break
    return a[:length] # Nicholas Stefan Georgescu, Jun 02 2022

Python

from math import isqrt
def A001468(n): return (n+1+isqrt(m:=5*(n+1)**2)>>1)-(n+isqrt(m-10*n-5)>>1) # Chai Wah Wu, Aug 25 2022

Formula

a(n) = [(n+1) tau] - [n tau], tau = (1 + sqrt 5)/2 = A001622, [] = floor function.

a(n) = A000201(n+1) - A000201(n) = A022342(n+1) - A022342(n), n >= 1; i.e., the first term discarded, this yields the first differences of A000201 and A022342. - M. F. Hasler, Oct 13 2017

Extensions

Rechecked by N. J. A. Sloane, Nov 07 2001