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%I #46 Jan 23 2024 08:41:23
%S 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,
%T 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,
%U 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
%N If n mod 3 = 0 then a(n) = a(n/3), otherwise a(n) = n mod 3.
%C A cubefree word. Start with 1, apply the morphisms 1 -> 121, 2 -> 122, take limit. See A080846 for another version.
%C Ultimate modulo 3: n-th digit of terms in "Ana sequence" (see A060032 for definition).
%C 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
%C Also equals A038502(n) mod 3.
%C Last nonzero digit in ternary representation of n. - _Franklin T. Adams-Watters_, Apr 01 2006
%C a(2*n) = length of n-th run of twos. - _Reinhard Zumkeller_, Mar 13 2015
%H Reinhard Zumkeller, <a href="/A060236/b060236.txt">Table of n, a(n) for n = 1..1000</a>
%H Jean Berstel and J. Karhumaki, <a href="http://www-igm.univ-mlv.fr/~berstel/Articles/2003TutorialCoWdec03.pdf">Combinatorics on words - a tutorial</a>, Bull. EATCS, #79 (2003), pp. 178-228.
%H <a href="/index/Fi#final">Index entries for sequences related to final digits of numbers</a>
%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%F a(3*n) = a(n), a(3*n + 1) = 1, a(3*n + 2) = 2. - _Michael Somos_, Jul 29 2009
%F a(n) = 1 + A080846(n). - _Joerg Arndt_, Jan 21 2013
%e a(10)=1 since 10=3^0*10 and 10 mod 3=1;
%e a(72)=2 since 24=3^3*8 and 8 mod 3=2.
%t Nest[ Flatten[ # /. {1 -> {1, 2, 1}, 2 -> {1, 2, 2}}] &, {1}, 5] (* _Robert G. Wilson v_, Mar 04 2005 *)
%t Table[Mod[n/3^IntegerExponent[n, 3], 3], {n, 1, 120}] (* _Clark Kimberling_, Oct 19 2016 *)
%t 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 *)
%o (PARI) a(n)=if(n<1, 0, n/3^valuation(n,3)%3) /* _Michael Somos_, Nov 10 2005 */
%o (Haskell) following _Franklin T. Adams-Watters_'s comment.
%o a060236 = head . dropWhile (== 0) . a030341_row
%o -- _Reinhard Zumkeller_, Mar 13 2015
%Y Cf. A026225 (indices of 1's), A026179 (indices of 2's).
%Y Cf. A060032 (concatenate 3^n terms).
%Y Cf. A030341, A007089.
%K easy,nonn
%O 1,2
%A _Henry Bottomley_, Mar 21 2001
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