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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

Index entries for sequences that are fixed points of mappings

Index entries for characteristic functions

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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Last modified January 19 21:35 EST 2018. Contains 297938 sequences.