A007484 a(n) = 3*a(n-1) + 2*a(n-2), with a(0)=2, a(1)=7. (Formerly M1767)

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

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

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).

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

Cf. A007455, A007481, A007483, 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

Keyword

easy,nonn,nice

Author

N. J. A. Sloane

Extensions

Definition edited by N. J. A. Sloane, Jul 30 2016

Status

approved