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A007482
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Number of subsequences of [ 1,...,2n ] in which each odd number has an even neighbor.
(Formerly M2893)
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34
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1, 3, 11, 39, 139, 495, 1763, 6279, 22363, 79647, 283667, 1010295, 3598219, 12815247, 45642179, 162557031, 578955451, 2061980415, 7343852147, 26155517271, 93154256107, 331773802863, 1181629920803, 4208437368135
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| The even neighbor must differ from the odd number by exactly one.
If we defined this sequence by the recurrence (a(n) = 3*a(n-1) + 2*a(n-2)) that it satisfies, we could prefix it with an initial 0.
a(n) equals term (1,2) in M^n, M = the 3x3 matrix [1,1,2; 1,0,1; 2,1,1]. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 12 2009]
a(n) equals term (2,2) in M^n, M = the 3x3 matrix [0,1,0; 1,3,1; 0,1,0]. [From Paul Barry (pbarry(AT)wit.ie), Sep 18 2009]
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 06 2010: (Start)
Starting with "1" = INVERT transform of A002605: (1, 2, 6, 16, 44,...).
Example: a(3) = 39 = (16, 6, 2, 1) dot (1, 1, 3, 11) = (16 + 6 + 6 + 11). (End)
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REFERENCES
| R. K. Guy, Moser, William O.J.: Numbers of subsequences without isolated odd members. Fibonacci Quarterly, 34, No. 2, 152-155 (1996). Math. Rev. 97d:11017.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 442
Index entries for sequences related to linear recurrences with constant coefficients, signature 3,2.
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FORMULA
| Let b(0)=1, b(k)=floor(b(k-1))+2/b(k-1); then, for n>0, b(n)=a(n)/a(n-1). - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 09 2002
The Hankel transform of this sequence is [1,2,0,0,0,0,0,0,0,...]. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 21 2007
G.f.: 1/(1-3x-2x^2). a(n)=3a(n-1)+2a(n-2). a(n)=(ap^(n+1)-am^(n+1))/(ap-am), ap := (3+sqrt(17))/2, am := (3-sqrt(17))/2.
a(n)=sum{k=0..floor(n/2), C(n-k, k)2^k*3^(n-2k)} - Paul Barry (pbarry(AT)wit.ie), Apr 23 2005
a(n)=Sum_{k, 0<=k<=n}A112906(n,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 21 2007
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MATHEMATICA
| a[n_]:=(MatrixPower[{{1, 4}, {1, 2}}, n].{{1}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 19 2010]
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PROG
| (Other) sage: [lucas_number1(n, 3, -2) for n in xrange(1, 25)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
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CROSSREFS
| Cf. A007455, A007481, A007483, A007484.
Row sums of triangle A073387.
Cf. A000045, A000129, A001045.
Cf. A002605 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 06 2010]
Sequence in context: A089579 A166336 A002783 * A134760 A132889 A149061
Adjacent sequences: A007479 A007480 A007481 * A007483 A007484 A007485
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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