Search: id:A003945
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%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 G.f.: (1+x)/(1-2*x).
%C A003945 Coordination sequence for infinite tree with valency 3.
%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 F. Faase, <a href="http://home.wxs.nl/~faase009/counting.html">Counting 
               Hamilton cycles in product graphs</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 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>
%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>
%F A003945 a(0) = 1; for n>0, a(n) = 3*2^(n-1).
%F A003945 a(n)=2a(n-1), n>1; a(0)=1, a(1)=3.
%F A003945 More generally, the g.f. (1+x)/(1-kx) 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
%F A003945 The g.f. (1+x)/(1-kx) produces the sequence with closed form (in PARI 
               notation) a(n)=(n>=0)*k^n+(n>=1)*k^(n-1). - Jaume Oliver i Lafont, 
               Dec 05 2009
%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, A001787, A001045, A161175.
%Y A003945 Generating functions of the form (1+x)/(1-kx) for k=1 to 12: A040000, 
               A003945, A003946, A003947, A003948, A003949, A003950, A003951, A003952
%Y A003945 Generating functions of the form (1+x)/(1-kx) for k=13 to 30: A170732 
               A170733 A170734 A170735 A170736 A170737 A170738 A170739 A170740 A170741 
               A170742 A170743 A170744 A170745 A170746 A170747 A170748
%Y A003945 Generating functions of the form (1+x)/(1-kx) 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
%Y A003945 Sequence in context: A169160 A122391 A169208 this_sequence A169256 A169304 
               A169352
%Y A003945 Adjacent sequences: A003942 A003943 A003944 this_sequence A003946 A003947 
               A003948
%K A003945 nonn,easy,new
%O A003945 0,2
%A A003945 N. J. A. Sloane (njas(AT)research.att.com).
%E A003945 Edited by N. J. A. Sloane (njas(AT)research.att.com), Dec 04 2009.

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