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

1,2

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

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

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

Example

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.

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 *)

Prog

(PARI) a(n)=if(n<1, 0, n/3^valuation(n, 3)%3) /* Michael Somos, Nov 10 2005 */
(Haskell)  following Franklin T. Adams-Watters's comment.
a060236 = head . dropWhile (== 0) . a030341_row
-- Reinhard Zumkeller, Mar 13 2015

Crossrefs

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

Cf. A060032 (concatenate 3^n terms).

Cf. A030341, A007089.

Sequence in context: A265209 A202340 A049705 * A006345 A122497 A350330

Adjacent sequences: A060233 A060234 A060235 * A060237 A060238 A060239

Keyword

easy,nonn

Author

Henry Bottomley, Mar 21 2001

Status

approved