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A003945
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G.f.: (1+x)/(1-2*x).
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77
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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; internal format)
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OFFSET
| 0,2
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COMMENTS
| Coordination sequence for infinite tree with valency 3.
Number of Hamiltonian cycles in K_3 X P_n.
Number of ternary squarefree words of length n.
Row sums of A029635. - Paul Barry (pbarry(AT)wit.ie), Jan 30 2005
Binomial transform is {1, 4, 13, 40, 121, 364, ...}, see A003462 . -Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 23 2005
Convolved with the Jacobsthal sequence A001045 = A001786: (1, 4, 12, 32, 80,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 23 2009]
Equals (n+1)-th row sums of triangle A161175 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 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)). [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Aug 17 2009]
Starting (1, 3, 6, 12,...) = INVERTi transform of A003688: (1, 4, 13, 43,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 05 2010]
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 15 2010: (Start)
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.
(End)
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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
C. Richard and U. Grimm, On the entropy and letter frequencies of ternary squarefree words
Index to divisibility sequences
Index entries for sequences related to linear recurrences with constant coefficients
Index entries for sequences related to trees
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FORMULA
| a(0) = 1; for n>0, a(n) = 3*2^(n-1).
a(n)=2a(n-1), n>1; a(0)=1, a(1)=3.
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
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
Binomial transform of A000034. a(n)=(3*2^n-0^n)/2 - Paul Barry (pbarry(AT)wit.ie), Apr 29 2003
a(n)=sum{k=0..n, (n+k)binomial(n, k)/n} - Paul Barry (pbarry(AT)wit.ie), Jan 30 2005
a(n) = Sum_{ 0<=k<=n } A029653(n, k)*x^k for x = 1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 10 2005
Binomial transform of A000034. Hankel transform is {1,-3,0,0,0,...}. - Paul Barry (pbarry(AT)wit.ie), Aug 29 2006
Row sums of triangle A133084 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 08 2007
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MAPLE
| k := 3; if n = 0 then 1 else k*(k-1)^(n-1); fi;
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MATHEMATICA
| Join[{1}, 3*2^Range[0, 60]] (* From Vladimir Joseph Stephan Orlovsky, June 09 2011 *)
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PROG
| (PARI) a(n)=if(n, 3<<n--, 1) \\ Charles R Greathouse IV, Jan 12 2012
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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
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Dec 04 2009.
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