A060236 If n mod 3 = 0 then a(n) = a(n/3), otherwise a(n) = n mod 3.

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

Offset: 1

Author

Henry Bottomley, Mar 21 2001

Keywords

• easy
• nonn

Comments

A cubefree word. Start with 1, apply the morphisms 1 -> 121, 2 -> 122, take limit. See A080846 for another version.

Ultimate modulo 3: n-th digit of terms in "Ana sequence" (see A060032 for definition).

Equals A005148(n) reduced mod 3. In "On a sequence Arising in Series for Pi" Morris Newman and Daniel Shanks conjectured that 3 never divides A005148(n) and D. Zagier proved it. - Benoit Cloitre, Jun 22 2002

Also equals A038502(n) mod 3.

Last nonzero digit in ternary representation of n. - Franklin T. Adams-Watters, Apr 01 2006

a(2*n) = length of n-th run of twos. - Reinhard Zumkeller, Mar 13 2015

Examples

			a(10)=1 since 10=3^0*10 and 10 mod 3=1;
a(72)=2 since 24=3^3*8 and 8 mod 3=2.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Jean Berstel and J. Karhumaki, Combinatorics on words - a tutorial, Bull. EATCS, #79 (2003), pp. 178-228.
• Index entries for sequences related to final digits of numbers
• Index entries for sequences that are fixed points of mappings

Crossrefs

Cf. A026225 (indices of 1's), A026179 (indices of 2's).

Cf. A060032 (concatenate 3^n terms).

Cf. A030341, A007089.

Programs

Haskell

following Franklin T. Adams-Watters's comment.
a060236 = head . dropWhile (== 0) . a030341_row
-- Reinhard Zumkeller, Mar 13 2015

Magma

[(Floor(n/3^Valuation(n, 3)) mod 3): n in [1..120]]; // G. C. Greubel, Nov 05 2024

Mathematica

Nest[ Flatten[ # /. {1 -> {1, 2, 1}, 2 -> {1, 2, 2}}] &, {1}, 5] (* Robert G. Wilson v, Mar 04 2005 *)
Table[Mod[n/3^IntegerExponent[n, 3], 3], {n, 1, 120}] (* Clark Kimberling, Oct 19 2016 *)
lnzd[m_]:=Module[{s=Split[m]},If[FreeQ[Last[s],0],s[[-1,1]],s[[-2,1]]]]; lnzd/@Table[IntegerDigits[n,3],{n,120}] (* Harvey P. Dale, Oct 19 2018 *)

PARI

a(n)=if(n<1, 0, n/3^valuation(n,3)%3) /* Michael Somos, Nov 10 2005 */

SageMath

[n/3^valuation(n, 3)%3 for n in range(1,121)] # G. C. Greubel, Nov 05 2024

Formula

a(3*n) = a(n), a(3*n + 1) = 1, a(3*n + 2) = 2. - Michael Somos, Jul 29 2009

a(n) = 1 + A080846(n). - Joerg Arndt, Jan 21 2013