%I A003945
%S A003945 1,3,6,12,24,48,96,192,384,768,1536,3072,6144,12288,24576,
%T A003945 49152,98304,196608,393216,786432,1572864,3145728,6291456,
%U A003945 12582912,25165824,50331648,100663296,201326592,402653184
%N A003945 Coordination sequence for infinite tree with valency 3: a(0) = 1; for 
               n>0, a(n) = 3*2^(n-1).
%C A003945 Number of Hamiltonian cycles in K_3 X P_n.
%C A003945 Number of ternary squarefree words of length n.
%C A003945 Row sums of A029635. - Paul Barry (pbarry(AT)wit.ie), Jan 30 2005
%C A003945 Binomial transform is {1, 4, 13, 40, 121, 364, ...}, see A003462 . -Philippe 
               DELEHAM (kolotoko(AT)wanadoo.fr), Jul 23 2005
%C A003945 Convolved with the Jacobsthal sequence A001045 = A001786: (1, 4, 12, 
               32, 80,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 23 
               2009]
%C A003945 Equals (n+1)-th row sums of triangle A161175 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jun 05 2009]
%C A003945 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)). [From Jaroslav 
               Krizek (jaroslav.krizek(AT)atlas.cz), Aug 17 2009]
%H A003945 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A003945 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%H A003945 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=151">
               Encyclopedia of Combinatorial Structures 151</a>
%H A003945 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=304">
               Encyclopedia of Combinatorial Structures 304</a>
%H A003945 F. Faase, <a href="http://home.wxs.nl/~faase009/counting.html">Counting 
               Hamilton cycles in product graphs</a>
%H A003945 C. Richard and U. Grimm, <a href="http://arXiv.org/abs/math.CO/0302302">
               On the entropy and letter frequencies of ternary square-free words</
               a>
%F A003945 G.f.: (1+x)/(1-2*x)
%F A003945 a(n)=2a(n-1), n>1; a(0)=1, a(1)=3.
%F A003945 Binomial transform of A000034. a(n)=(3*2^n-0^n)/2 - Paul Barry (pbarry(AT)wit.ie), 
               Apr 29 2003
%F A003945 a(n)=sum{k=0..n, (n+k)binomial(n, k)/n} - Paul Barry (pbarry(AT)wit.ie), 
               Jan 30 2005
%F A003945 a(n) = Sum_{ 0<=k<=n } A029653(n, k)*x^k for x = 1 . - Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Jul 10 2005
%F A003945 Binomial transform of A000034. Hankel transform is {1,-3,0,0,0,...}. 
               - Paul Barry (pbarry(AT)wit.ie), Aug 29 2006
%F A003945 Row sums of triangle A133084 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Sep 08 2007
%F A003945 a(0) = 1, a(n) = 3*2^(n-1) = 2^n + 2^(n-1) for n >= 1. [From Jaroslav 
               Krizek (jaroslav.krizek(AT)atlas.cz), Aug 17 2009]
%p A003945 k := 3; if n = 0 then 1 else k*(k-1)^(n-1); fi;
%p A003945 with(combinat):a:=n->stirling2(n,2)-stirling2(n-2,2): seq(a(n), n=2..30); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007
%p A003945 with(finance):seq(floor(futurevalue(3,1,n)), n=-1..27);# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Mar 24 2009]
%o A003945 (Other) SAGE:[lucas_number1(n, 2, 0)+lucas_number1(n+1, 2, 0)for n in 
               xrange(0, 29)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 12 2009]
%Y A003945 Essentially same as A007283 (3*2^n) and A042950.
%Y A003945 Cf. A133084.
%Y A003945 A001787, A001045 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 23 
               2009]
%Y A003945 A161175 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 05 2009]
%Y A003945 Sequence in context: A115805 A046944 A122391 this_sequence A007283 A049942 
               A099844
%Y A003945 Adjacent sequences: A003942 A003943 A003944 this_sequence A003946 A003947 
               A003948
%K A003945 nonn,easy
%O A003945 0,2
%A A003945 N. J. A. Sloane (njas(AT)research.att.com).