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A007484
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a(n) = 3*a(n-1) + 2*a(n-2), with a(0)=2, a(1)=7.
(Formerly M1767)
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8
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2, 7, 25, 89, 317, 1129, 4021, 14321, 51005, 181657, 646981, 2304257, 8206733, 29228713, 104099605, 370756241, 1320467933, 4702916281, 16749684709, 59654886689, 212464029485, 756701861833, 2695033644469, 9598504657073, 34185581260157, 121753753094617
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OFFSET
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0,1
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COMMENTS
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Number of subsequences of [1,...,2n+1] in which each even number has an odd neighbor.
Same as Pisot sequence E(2,7) (see A008776).
8*a(n) = A007482(n+2) + A007483(n+1) (conjectured, see A104934 for related formula). - Creighton Dement, Apr 15 2005
Conjecture verified using generating functions. - Robert Israel, Jul 12 2018
a(n) = sum of the elements of the matrix M^n, where M = {{1, 2}, {2, 2}}. - Griffin N. Macris, Mar 25 2016
a(3) = 25 is the only composite among the first 8 terms, but then the density of primes decreases, dropping below 50% at the 27th term. - M. F. Hasler, Jul 12 2018
a(n) is also the number of dominating sets in the (2n+1)-triangular snake graph for n > 0. - Eric W. Weisstein, Jun 09 2019
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
R. K. Guy and W. O. J. Moser, Numbers of subsequences without isolated odd members, Fibonacci Quarterly 34:2 (1996), pp. 152-155.
Eric Weisstein's World of Mathematics, Dominating Set
Eric Weisstein's World of Mathematics, Triangular Snake Graph
Index entries for linear recurrences with constant coefficients, signature (3, 2).
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FORMULA
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a(n) = (3/2 + sqrt(17)/2)^n - (4/17)*sqrt(17)*(3/2 - sqrt(17)/2)^n + (4/17)*(3/2 + sqrt(17)/2)^n*sqrt(17) + (3/2 - sqrt(17)/2)^n, with n >= 0. - Paolo P. Lava, Jun 11 2008
a(n) = nearest integer to (and converges rapidly to) (1+4/sqrt(17))*((3+sqrt(17))/2)^n. - N. J. A. Sloane, Jul 30 2016
If p[i] = Fibonacci(i+2) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. - Milan Janjic, May 08 2010
G.f.: (2 + x)/(1 - 3*x - 2*x^2). - M. F. Hasler, Jul 12 2018
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MAPLE
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A007484 := proc(n) option remember; if n=0 then 2; elif n=1 then 7; else 3*A007484(n-1)+2*A007484(n-2); fi; end;
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MATHEMATICA
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LinearRecurrence[{3, 2}, {2, 7}, 40] (* Harvey P. Dale, Apr 24 2012 *)
Table[(2^-n ((3 - Sqrt[17])^n (-4 + Sqrt[17]) + (3 + Sqrt[17])^n (4 + Sqrt[17])))/Sqrt[17], {n, 0, 20}] // Expand (* Eric W. Weisstein, Jun 09 2019 *)
CoefficientList[Series[(-2 - x)/(-1 + 3 x + 2 x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 09 2019 *)
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PROG
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(Haskell)
a007484 n = a007484_list !! n
a007484_list = 2 : 7 : zipWith (+)
(map (* 3) $ tail a007484_list) (map (* 2) a007484_list)
-- Reinhard Zumkeller, Nov 02 2015
(PARI) a(n)=([0, 1; 2, 3]^n*[2; 7])[1, 1] \\ Charles R Greathouse IV, Mar 25 2016
(PARI) A007484_vec(N)=Vec((2+x)/(1-3*x-2*x^2)+O(x^n)) \\ M. F. Hasler, Jul 12 2018
(MAGMA) A007484:=[2, 7]; [n le 2 select A007484[n] else 3*Self(n-1)+2*Self(n-2): n in [1..40]]; // Wesley Ivan Hurt, Jan 24 2017
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CROSSREFS
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Cf. A007455, A007481, A007484.
See A008776 for definitions of Pisot sequences.
Sequence in context: A289598 A030017 A131430 * A070859 A048576 A018907
Adjacent sequences: A007481 A007482 A007483 * A007485 A007486 A007487
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KEYWORD
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easy,nonn,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Definition edited by N. J. A. Sloane, Jul 30 2016
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STATUS
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approved
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