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 A039966 a(0) = 1, a(3n+2) = 0, a(3n) = a(3n+1) = a(n). 30
 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Number of partitions of n into distinct powers of 3. Trajectory of 1 under the morphism: 1 -> 110, 0 -> 000. Thus 1 -> 110 ->110110000 -> 110110000110110000000000000 -> ... - Philippe Deléham, Jul 09 2005 Also, an example of a d-perfect sequence. This is a composite of two earlier sequences contributed at different times by N. J. A. Sloane and by Reinhard Zumkeller, Mar 05 2005. Christian G. Bower extended them and found that they agreed for at least 512 terms. The proof that they were identical was found by Ralf Stephan, Jun 13 2005, based on the fact that they were both 3-regular sequences. LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions, in Sequences and their Applications, C. Ding, T. Helleseth, and H. Niederreiter, eds., Proceedings of SETA'98 (Singapore, 1998), 308-317, 1999. N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS FORMULA a(0)=1, a(1)=0, a(n) = b(n-2), where b [A-number?] is the sequence defined by b(0) = 1, b(3n+2) = 0, b(3n) = b(3n+1) = b(n). - Ralf Stephan a(n) = A005043(n-1) mod 3. - Christian G. Bower, Jun 12 2005 Properties: 0 <= a(n) <= 1, a(A074940(n)) = 0, a(A005836(n)) = 1; A104406(n) = Sum(a(k): 1<=k<=n). - Reinhard Zumkeller, Mar 05 2005 Euler transform of sequence b(n) where b(3^k)=1, b(2*3^k)=-1 and zero otherwise. - Michael Somos, Jul 15 2005 G.f. A(x) satisfies A(x)=(1+x)A(x^3). - Michael Somos, Jul 15 2005 G.f.: Product{k>=0} 1+x^(3^k). Exponents give A005836. a(n) = A002426(n) mod 3. - John M. Campbell, Aug 24 2011 a(n) = A000275(n) mod 3. - John M. Campbell, Jul 08 2016 EXAMPLE The triplets of elements (a(3k), a(3k+1), a(3k+2)) are (1,1,0) if a(k) = 1 and (0,0,0) if a(k) = 0.  So since a(2) = 0, a(6) = a(7) = a(8) = 0, and since a(3) = 1, a(9) = a(10) = 1 and a(11) = 0. - Michael B. Porter, Jul 11 2016 MAPLE a := proc(n) option remember; if n <= 1 then RETURN(1) end if; if n = 2 then RETURN(0) end if; if n mod 3 = 2 then RETURN(0) end if; if n mod 3 = 0 then RETURN(a(1/3*n)) end if; if n mod 3 = 1 then RETURN(a(1/3*n - 1/3)) end if end proc; # Ralf Stephan, Jun 13 2005 MATHEMATICA (* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) s = Rest[ Sort[ Plus @@@ Table[UnrankSubset[n, Table[3^i, {i, 0, 4}]], {n, 32}]]]; Table[ If[ Position[s, n] == {}, 0, 1], {n, 105}] (* Robert G. Wilson v, Jun 14 2005 *) CoefficientList[Series[Product[(1 + x^(3^k)), {k, 0, 5}], {x, 0, 111}], x] (* or *) Nest[ Flatten[ # /. {0 -> {0, 0, 0}, 1 -> {1, 1, 0}}] &, {1}, 5] (* Robert G. Wilson v, Mar 29 2006 *) Nest[ Join[#, #, 0 #] &, {1}, 5] (* Robert G. Wilson v, Jul 27 2014 *) PROG (PARI) {a(n)=local(A, m); if(n<0, 0, m=1; A=1+O(x); while(m<=n, m*=3; A=(1+x)*subst(A, x, x^3)); polcoeff(A, n))} /* Michael Somos, Jul 15 2005 */ (Haskell) a039966 n = fromEnum (n < 2 || m < 2 && a039966 n' == 1)    where (n', m) = divMod n 3 -- Reinhard Zumkeller, Sep 29 2011 CROSSREFS For generating functions Prod_{k>=0} (1+a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674. Cf. A062051, A000009, A000244, A004642, A005836. Sequence in context: A077050 A128432 A195198 * A089451 A145099 A070886 Adjacent sequences:  A039963 A039964 A039965 * A039967 A039968 A039969 KEYWORD nonn AUTHOR N. J. A. Sloane, Dec 11 1999 EXTENSIONS Entry revised Jun 30 2005 Offset corrected by John M. Campbell, Aug 24 2011 STATUS approved

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