%I A005836 M2353
%S A005836 0,1,3,4,9,10,12,13,27,28,30,31,36,37,39,40,81,82,84,85,90,91,93,94,
%T A005836 108,109,111,112,117,118,120,121,243,244,246,247,252,253,255,256,270,
%U A005836 271,273,274,279,280,282,283,324,325,327,328,333,334,336,337,351,352
%N A005836 Numbers n such that base 3 representation contains no 2.
%C A005836 3 does not divide binomial(2s,s) if and only if s is a member of this 
               sequence, where binomial(2s,s)= A000984(s) are the central binomial 
               coefficients.
%C A005836 This is the "earliest" sequence obtained among nonnegative numbers by 
               forbidding arithmetic subsequences of length 3 - Robert Craigen (craigenr(AT)cc.umanitoba.ca), 
               Jan 29 2001
%C A005836 Complement of A074940. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Mar 23 2003
%C A005836 Sums of distinct powers of 3. - Ralf Stephan (ralf(AT)ark.in-berlin.de), 
               Apr 27 2003
%C A005836 n such that central trinomial coefficient A002426(n) == 1 (mod 3). - 
               Emeric Deutsch and Bruce Sagan, Dec 04 2003
%C A005836 A039966(a(n)+1) = 1; A104406(n) = number of terms <= n.
%C A005836 Subsequence of A125292; A125291(a(n)) = 1 for n>1. - Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), Nov 26 2006
%C A005836 Also final value of n-1 written in base 2 and then read in base 3 and 
               with finally the result translated in base 10. - Philippe LALLOUET 
               (philip.lallouet(AT)wanadoo.fr), Jun 23 2007
%C A005836 A081603(a(n)) = 0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Mar 02 2008
%C A005836 Subsequence of A154314. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jan 07 2009]
%C A005836 a(n) modulo 2 is the Thue-Morse sequence A010060. [From Dennis Tseng 
               (Dtseng(AT)cinci.rr.com), Jul 16 2009]
%D A005836 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005836 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical 
               Computer Sci., 98 (1992), 163-197.
%D A005836 R. K. Guy, Unsolved Problems in Number Theory, E10.
%H A005836 T. D. Noe, <a href="b005836.txt">Table of n, a(n) for n=1..1024</a>
%H A005836 J.-P. Allouche and J. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/
               Papers/as0.ps">The ring of k-regular sequences</a>, Theoretical Computer 
               Sci., 98 (1992), 163-197.
%H A005836 J.-P. Allouche, J. Shallit and G. Skordev, <a href="http://www.lri.fr/
               ~allouche/kimb.ps">Self-generating sets, integers with missing blocks 
               and substitutions</a>, Discrete Math. 292 (2005) 1-15.
%H A005836 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Some Properties of a Certain Nonaveraging Sequence</a>, J. Integer 
               Sequences, Vol. 2, 1999, #4.
%H A005836 A. M. Odlyzko and R. P. Stanley, Some curious sequences constructed with 
               the greedy algorithm, 1978, remark 1 (<a href="http://www.dtc.umn.edu/
               ~odlyzko/unpublished/greedy.sequence.pdf">PDF</a>, <a href="http:/
               /www.dtc.umn.edu/~odlyzko/unpublished/greedy.sequence.ps">PS</a>, 
               <a href="http://www.dtc.umn.edu/~odlyzko/unpublished/greedy.sequence.tex">
               TeX</a>).
%H A005836 P. Pollack, <a href="http://www.math.dartmouth.edu/~ppollack/notes.pdf">
               Analytic and Combinatorial Number Theory</a> Course Notes, p. 228.
%H A005836 R. Stephan, <a href="http://arXiv.org/abs/math.CO/0307027">Divide-and-conquer 
               generating functions. I. Elementary sequences</a>
%H A005836 R. Stephan, <a href="somedcgf.html">Some divide-and-conquer sequences 
               ...</a>
%H A005836 R. Stephan, <a href="a079944.ps">Table of generating functions</a>
%H A005836 Z. Sunic, <a href="http://arXiv.org/abs/math.CO/0612080">Tree morphisms, 
               transducers and integer sequences</a>
%H A005836 B. Vasic, K. Pedagani and M. Ivkovic, <a href="http://ieeexplore.ieee.org/
               xpls/abs_all.jsp?arnumber=1327838">High-rate girth-eight low-density 
               parity-check codes on rectangular integer lattices</a>, IEEE Transactions 
               on Communications, Vol. 52, Issue 8 (2004), 1248-1252.
%H A005836 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CentralBinomialCoefficient.html">Central Binomial Coefficient</a>
%F A005836 a(n+1) = sum( b(k)* 3^k ) for k=0..m and n = sum( b(k)* 2^k )
%F A005836 a(2n+1)=3a(n+1), a(2n+2)=a(2n+1)+1, a(0)=0.
%F A005836 a(n+1)=3*a(floor(n/2))+n-2*floor(n/2) - Ralf Stephan (ralf(AT)ark.in-berlin.de), 
               Apr 27 2003
%F A005836 G.f. x/(1-x) * Sum(k>=0, 3^k*x^2^k/(1+x^2^k)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), 
               Apr 27 2003
%F A005836 n such that the coefficient of x^n is > 0 in prod (k>=0, 1+x^(3^k)) - 
               Benoit Cloitre (benoit7848c(AT)orange.fr), Jul 29 2003
%F A005836 a(n) = Sum_{k = 1..n-1} (1 + 3^A007814(k)) / 2 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Jul 09 2005
%F A005836 If the offset were changed to zero, then: a(0)=0, a(n+1)=f(a(n)+1,f(a(n)+1) 
               where f(x,y) = if x<3 and x<>2 then y else if x mod 3 = 2 then f(y+1, 
               y+1) else f(floor(x/3),y). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Mar 02 2008
%e A005836 a(6) = 12 because 6 = 0*2^0 +1*2^1 +1*2^2 = 2+4 and 12 = 0*3^0 +1*3^1 
               +1*3^2 = 3+9.
%t A005836 Table[FromDigits[IntegerDigits[k, 2], 3], {k, 60}]
%o A005836 (PARI) a(n)=n--;if(n<1,0,if(n%2,a(n-1)+1,3*a(n/2)))
%o A005836 (PARI) a(n)=n--;if(n<1,0,3*a(floor(n/2))+n-2*floor(n/2))
%Y A005836 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.
%Y A005836 a(n) = A005823(n)/2; a(n) = A003278(n)-1 = A033159(n)-2 = A033162(n)-3.
%Y A005836 Cf. A005823, A032924, A054591, A007089, A081603, A081611, A000695, A007088, 
               A033042-A033052, A074940, A083096. A002426.
%Y A005836 Cf. A003278, A004793, A055246, A062548, A081601, A089118.
%Y A005836 Row 3 of array A104257.
%Y A005836 Sequence in context: A010388 A010400 A010439 this_sequence A054591 A121153 
               A059985
%Y A005836 Adjacent sequences: A005833 A005834 A005835 this_sequence A005837 A005838 
               A005839
%K A005836 nonn,nice,easy
%O A005836 1,3
%A A005836 N. J. A. Sloane (njas(AT)research.att.com), Jeffrey Shallit
%E A005836 More terms from Emeric Deutsch and Bruce Sagan, Dec 04 2003
%E A005836 Offset corrected by N. J. A. Sloane (njas(AT)research.att.com), Mar 02 
               2008.
%E A005836 Edited by the Associate Editors of the OEIS, Apr 07 2009