A003945 Expansion of g.f. (1+x)/(1-2*x).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

INVERTi transform of A003688. - 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
• Yasemin Alp and E. Gokcen Kocer, Exponential Almost-Riordan Arrays, Results Math. (2024) Vol. 79, 173.
• F. Faase, Counting Hamiltonian cycles in product graphs
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 151
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 304
• Markus Kuba and Alois Panholzer, 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, arXiv:math/0302302 [math.CO], 2003.
• Index to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (2).
• Index entries for sequences related to trees

Crossrefs

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

Cf. 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.

Programs

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 *)

PARI

a(n)=if(n,3<Charles R Greathouse IV, Jan 12 2012

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_{k=0..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

a(0) = 1, a(n) = 2 + Sum_{k=0..n-1} a(k) for n >= 1. - Joerg Arndt, Aug 15 2012

a(n) = 2^n + floor(2^(n-1)). - Martin Grymel, Oct 17 2012

E.g.f.: (3*exp(2*x) - 1)/2. - Stefano Spezia, Jan 31 2023

Extensions

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