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 A003945 Expansion of g.f. (1+x)/(1-2*x). 202
 1, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944, 12884901888 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Coordination sequence for infinite tree with valency 3. Number of Hamiltonian cycles in K_3 X P_n. Number of ternary words of length n avoiding aa, bb, cc. For n>0, row sums of A029635. - Paul Barry, Jan 30 2005 Binomial transform is {1, 4, 13, 40, 121, 364, ...}, see A003462 . -Philippe Deléham, Jul 23 2005 Convolved with the Jacobsthal sequence A001045 = A001786: (1, 4, 12, 32, 80,...). - Gary W. Adamson, May 23 2009 Equals (n+1)-th row sums of triangle A161175. - Gary W. Adamson, Jun 05 2009 a(n) written in base 2: a(0) = 1, a(n) for n >= 1: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, (n-1) times 0 (see A003953(n)). - Jaroslav Krizek, Aug 17 2009 Starting (1, 3, 6, 12,...) = INVERTi transform of A003688: (1, 4, 13, 43,...). - Gary W. Adamson, Aug 05 2010 An elephant sequence, see A175655. For the central square four A vectors, with decimal values 42, 138, 162 and 168, lead to this sequence. For the corner squares these vectors lead to the companion sequence A083329. - Johannes W. Meijer, Aug 15 2010 A216022(a(n)) != 2 and A216059(a(n)) != 3. - Reinhard Zumkeller, Sep 01 2012 Number of length-n strings of 3 letters with no two adjacent letters identical. The general case (strings of r letters) is the sequence with g.f. (1+x)/(1-(r-1)*x). - Joerg Arndt, Oct 11 2012 Sums of pairs of rows of Pascal's Triangle A007318, T(2n,k)+T(2n+1,k); sum(n>=1, A000290(n)/a(n) ) = 4. - John Molokach, Sep 26 2013 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 151 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 304 Kuba, Markus; Panholzer, Alois, Enumeration formulas for pattern restricted Stirling permutations, Discrete Math. 312 (2012), no. 21, 3179--3194. MR2957938. - From N. J. A. Sloane, Sep 25 2012 C. Richard and U. Grimm, On the entropy and letter frequencies of ternary squarefree words Index entries for linear recurrences with constant coefficients, signature (2). FORMULA a(0) = 1; for n>0, a(n) = 3*2^(n-1). a(n) = 2*a(n-1), n>1; a(0)=1, a(1)=3. More generally, the g.f. (1+x)/(1-k*x) produces the sequence [1, 1 + k, (1 + k)*k, (1 + k)*k^2,... (1+k)*k^(n-1),...], with a(0) = 1, a(n) = (1+k)*k^(n-1) for n >= 1. Also a(n+1) = k*a(n) for n >= 1. - Zak Seidov and N. J. A. Sloane, Dec 05 2009 The g.f. (1+x)/(1-k*x) produces the sequence with closed form (in PARI notation) a(n)=(n>=0)*k^n+(n>=1)*k^(n-1). - Jaume Oliver Lafont, Dec 05 2009 Binomial transform of A000034. a(n)=(3*2^n-0^n)/2. - Paul Barry, Apr 29 2003 a(n)=sum{k=0..n, (n+k)binomial(n, k)/n}. - Paul Barry, Jan 30 2005 a(n) = Sum_{ 0<=k<=n } A029653(n, k)*x^k for x = 1. - Philippe Deléham, Jul 10 2005 Binomial transform of A000034. Hankel transform is {1,-3,0,0,0,...}. - Paul Barry, Aug 29 2006 Row sums of triangle A133084. - Gary W. Adamson, Sep 08 2007 a(0) = 1, a(n) = sum(k=0..n-1, a(k) ) + 2 for n>=1. - Joerg Arndt, Aug 15 2012 a(n) = 2^{n}+floor(2^{n-1}). - Martin Grymel, Oct 17 2012 MAPLE k := 3; if n = 0 then 1 else k*(k-1)^(n-1); fi; MATHEMATICA Join[{1}, 3*2^Range[0, 60]] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *) Table[2^n+Floor[2^(n-1)], {n, 0, 30}] (* Martin Grymel, Oct 17 2012 *) CoefficientList[Series[(1+x)/(1-2x), {x, 0, 40}], x] (* or *) LinearRecurrence[ {2}, {1, 3}, 40] (* Harvey P. Dale, May 04 2017 *) PROG (PARI) a(n)=if(n, 3<

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Last modified October 20 15:29 EDT 2019. Contains 328267 sequences. (Running on oeis4.)