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A007484 a(n) = 3*a(n-1) + 2*a(n-2), with a(0)=2, a(1)=7.
(Formerly M1767)
8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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
FORMULA
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
From G. C. Greubel, Jul 18 2021: (Start)
a(n) = (i*sqrt(2))^(n-1)*( i*2*sqrt(2)*ChebyshevU(n, -3*i/(2*sqrt(2))) + ChebyshevU(n-1, -3*i/(2*sqrt(2))) ).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*((7*n-8*k)/(n-k))*2^k*3^(n-2*k-1) with a(0) = 2. (End)
If we extend the definition of A007483(m) to negative m by using the recurrence, then a(n) = A007483(-3-n)*(-2)^n holds for all n in Z. - Michael Somos, Jul 19 2021
EXAMPLE
G.f. = 2 + 7*x + 25*x^2 + 89*x^3 + 317*x^4 + 1129*x^5 + ... - Michael Somos, Jul 19 2021
MAPLE
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;
MATHEMATICA
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 -3x -2x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 09 2019 *)
a[ n_] := MatrixPower[{{1, 2}, {2, 2}}, n]//Flatten//Total; (* Michael Somos, Jul 19 2021 *)
PROG
(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
(Sage) [(i*sqrt(2))^(n-1)*( i*2*sqrt(2)*chebyshev_U(n, -3*i/(2*sqrt(2))) + chebyshev_U(n-1, -3*i/(2*sqrt(2))) ) for n in (0..30)] # G. C. Greubel, Jul 18 2021
CROSSREFS
See A008776 for definitions of Pisot sequences.
Sequence in context: A289598 A030017 A131430 * A070859 A048576 A018907
KEYWORD
easy,nonn,nice
AUTHOR
EXTENSIONS
Definition edited by N. J. A. Sloane, Jul 30 2016
STATUS
approved

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Last modified February 27 07:37 EST 2024. Contains 370367 sequences. (Running on oeis4.)