A008351 a(n) is the concatenation of a(n-1) and a(n-2) with a(1)=1, a(2)=2.

1, 2, 21, 212, 21221, 21221212, 2122121221221, 212212122122121221212, 2122121221221212212122122121221221, 2122121221221212212122122121221221212212122122121221212

Offset

1,2

Comments

A "non-commutative Fibonacci" sequence. Often written as: a, b, ba, bab, babba, babbabab, babbababbabba, babbababbabbababbabab, ...

Converges in the appropriate topology. - Dylan Thurston, Jan 28 2005

Do a web search on babbababbabbababbabab to get further links.

a(n) has Fibonacci(n) digits d_i where 1 <= i <= n and n > 2. If i is in A001950 then d_i = 1, otherwise it is 2 [Stolarsky]. - David A. Corneth, May 14 2017

References

D. E. Knuth, "The Art of Programming", Volume 1, "Fundamental Algorithms", third edition, problem 36 on page 86.

Links

Table of n, a(n) for n=1..10.

K. B. Stolarsky, Beatty sequences, continued fractions, and certain shift operators, Canadian Math. Bull. 19 (1976) pp. 473-482.

Wikipedia, Lindenmayer system

Mathematica

a[1] = 1; a[2] = 2; a[n_] := 10^Floor[ Log[10, a[n - 2]] +1]*a[n - 1] + a[n - 2] (* Robert G. Wilson v, Jan 26 2006 *)

Prog

(PARI) a(n) = if (n<=2, n, eval(concat(Str(a(n-1)), Str(a(n-2))))); \\ Michel Marcus, May 14 2017
(PARI) a(n) = {if(n<=2, return(n));
my(v=vector(fibonacci(n), i, 2), phi2 = (3+sqrt(5))/2, b = vector(fibonacci(n-2), i, (i*(sqrt(5)+3)/2))\1); for(i=1, fibonacci(n-2), v[(i*(3+sqrt(5))/2)\1] = 1); sum(i=1, #v, 10^(#v-i) * v[i])}
a(n) = my(v=vector(n)); if(n <= 2, return(n)); v[1] = 1; v[2] = 2; for(i=3, n, v[i]=eval(concat(Str(v[i-1]), Str(v[i-2])))); v[#v] \\ David A. Corneth, May 14 2017

Crossrefs

See A008352 for another version.

Cf. A014675: 1->2, 2->21.

Cf. A001950.

Sequence in context: A304272 A037575 A305659 * A037743 A037638 A131698

Adjacent sequences: A008348 A008349 A008350 * A008352 A008353 A008354

Keyword

nonn,base

Author

N. J. A. Sloane and J. H. Conway

Extensions

Title clarified by Chai Wah Wu, Mar 17 2021

Status

approved