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A007483
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Number of subsequences of [ 1,...,2n+1 ] in which each odd number has an even neighbor.
(Formerly M3875)
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12
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1, 5, 17, 61, 217, 773, 2753, 9805, 34921, 124373, 442961, 1577629, 5618809, 20011685, 71272673, 253841389, 904069513, 3219891317, 11467812977, 40843221565, 145465290649, 518082315077, 1845177526529, 6571697209741
(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.
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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
| Vincenzo Librandi, Table of n, a(n) for n = 0..500
A. Burstein, S. Kitaev and T. Mansour, Independent sets in certain classes of (almost) regular graphs
Index to sequences with linear recurrences with constant coefficients, signature (3,2).
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FORMULA
| G.f.: (1+2*x)/(1-3*x-2*x^2).
a(n)=3*a(n-1)+2*a(n-2).
This sequence seems to be generated by the floretion - 0.5'i + 0.5j' + 0.25'ii' + 0.25'jj' - 0.75'kk' + 'ij' - 'ji' - 0.5'jk' - 0.5'ki' - 0.75e ("emseq") - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Nov 25 2004
a(n)=(3/2+sqrt(17)/2)^n*(1/2+7*sqrt(17)/34)+(1/2-7*sqrt(17)/34)*(3/2-sqrt(17)/2)^n - Paul Barry, Dec 08 2004
a(n-1) = Sum_{k, 0<=k<=n}2^(n-k)*A122542(n,k), n>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 08 2006
a(n) = upper left term in the 2 X 2 matrix [1,2; 2,2]^(n+1). Also [a(n), a(n+1)] = the 2 X 2 matrix [0,1; 2,3]^(n+1) * [1,1]. Example: [0,1; 2,3]^4 * [1,1] = [61, 217]. - Gary W. Adamson, Mar 16 2008
Also, for n>=2, a(n)=[1,2;2,2]^(n-1)*[1,2]*[1,2], [From John M. Campbell, Jul 09 2011]
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MATHEMATICA
| a[n_]:=(MatrixPower[{{2, 2}, {2, 1}}, n].{{2}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] [From Vladimir Orlovsky, Feb 20 2010]
f[n_]:=2/(n+3); x=2; Table[x=f[x]; Numerator[x], {n, 0, 5!}]/2 [From Vladimir Orlovsky, Mar 12 2010]
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PROG
| (MAGMA) [Floor((3/2+Sqrt(17)/2)^n*(1/2+7*Sqrt(17)/34)+(1/2-7*Sqrt(17)/34)*(3/2-Sqrt(17)/2)^n)+1: n in [0..30]]; // Vincenzo Librandi, Jul 09 2011
(PARI) a(n)=([1, 2; 2, 2]^n*[1, 2]~*[1, 2])[1, 1] \\ Charles R Greathouse IV, Jul 10 2011
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CROSSREFS
| Cf. A007482.
Sequence in context: A026619 A142956 A192146 * A149662 A149663 A149664
Adjacent sequences: A007480 A007481 A007482 * A007484 A007485 A007486
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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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