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A003945 G.f.: (1+x)/(1-2*x). 162
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[5] 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

F. Faase, Counting Hamilton cycles in product graphs

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 to divisibility sequences

Index entries for linear recurrences with constant coefficients, signature (2).

Index entries for sequences related to trees

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<<n--, 1) \\ Charles R Greathouse IV, Jan 12 2012

CROSSREFS

Essentially same as A007283 (3*2^n) and A042950.

Cf. A133084, A001787, A001045, A161175.

Generating functions of the form (1+x)/(1-k*x) for k=1 to 12: A040000, A003945, A003946, A003947, A003948, A003949, A003950, A003951, A003952.

Generating functions of the form (1+x)/(1-k*x) for k=13 to 30: A170732 A170733 A170734 A170735 A170736 A170737 A170738 A170739 A170740 A170741 A170742 A170743 A170744 A170745 A170746 A170747 A170748.

Generating functions of the form (1+x)/(1-k*x) for k=31 to 50: A170749 A170750 A170751 A170752 A170753 A170754 A170755 A170756 A170757 A170758 A170759 A170760 A170761 A170762 A170763 A170764 A170765 A170766 A170767 A170768 A170769.

Cf. A003688.

Sequence in context: A170636 A170684 * A007283 A049942 A200463 A099844

Adjacent sequences:  A003942 A003943 A003944 * A003946 A003947 A003948

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by N. J. A. Sloane, Dec 04 2009

STATUS

approved

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Last modified August 17 01:56 EDT 2017. Contains 290631 sequences.