A003683 a(n) = 2^(n-1)*(2^n - (-1)^n)/3.

0, 1, 2, 12, 40, 176, 672, 2752, 10880, 43776, 174592, 699392, 2795520, 11186176, 44736512, 178962432, 715816960, 2863333376, 11453202432, 45813071872, 183251763200, 733008101376, 2932030308352, 11728125427712

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) = A001045(n) * A011782(n). - Paul Barry, May 20 2003

The sequence 1,2,12,... is the binomial transform of (1, 1, 9, 9, 81, 81, ...) = 2*3^n/3 + (-3)^n/3. - Paul Barry, Jul 17 2003

Form a graph whose adjacency matrix is the tensor product of that of C_3 and [1,1;1,1]. a(n) counts walks of length n between any pair of adjacent nodes. A054881(n) counts closed walks of length n at a node.

Arises in connection with merit factor of the GRS sequences - see Hoeholdt et al.

2*a(n) = the constant term of the reduction by x^2->x+2 of the polynomial p(n,x) = ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+2); see A192382. For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232. - Clark Kimberling, Jun 30 2011

Apparently a(n+1) is the number of 3D tilings of a 2 X 2 X n room with bricks of 1 X 2 X 2 shape. - R. J. Mathar, Dec 06 2013

The ratio a(n+1)/a(n) converges to 4 as n approaches infinity. - Felix P. Muga II, Mar 10 2014

References

• M. Gardner, Riddles of the Sphinx, New Mathematical Library, M.A.A., 1987, p. 145. Math. Rev. 89i:00015.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• T. Hoeholdt, H. E. Jensen, and J. Justesen, Aperiodic correlations and the merit factor of a class of binary sequences, IEEE Trans. Inform. Theory, 13 (1985), 549-552
• R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 7.
• F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
• Eric Weisstein's World of Mathematics, Octahedral Graph
• Index entries for linear recurrences with constant coefficients, signature (2,8).

Crossrefs

Cf. A001045, A003674, A011782, A054881, A192232, A192382, A238801.

Programs

Magma

[2^(n-1)*(2^n - (-1)^n)/3: n in [0..30]]; // Vincenzo Librandi, Aug 19 2011

Maple

A003683:=n->2^(n-1)*(2^n - (-1)^n)/3; seq(A003683(n), n=0..50); # Wesley Ivan Hurt, Dec 06 2013

Mathematica

Table[2^(n-1) (2^n-(-1)^n)/3,{n,0,30}] (* or *) LinearRecurrence[{2,8},{0,1},30] (* Harvey P. Dale, Sep 15 2013 *)

PARI

a(n)=if(n<0,0,2^(n-1)*(2^n-(-1)^n)/3)

PARI

a(n)=(2^n-(-1)^n)<<(n-1)/3 \\ Charles R Greathouse IV, Apr 17 2012

Sage

[lucas_number1(n,2,-8) for n in range(0, 24)] # Zerinvary Lajos, Apr 22 2009

Formula

a(n) = A003674(n)/3.

a(n) = 2*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1. - Barry E. Williams, Jan 04 2000

G.f.: x/((1+2*x)*(1-4*x)).

a(n) = ((1+3)^n-(1-3)^n)/6. - Paul Barry, May 14 2003

a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k+1)*9^k. - Paul Barry, May 20 2003

E.g.f.: exp(x)*sinh(3*x)/3. - Paul Barry, Jul 09 2003

a(n+1) = 2^n*A001045(n+1). - R. J. Mathar, Jul 08 2009

a(n+1) = Sum_{k=0..n} A238801(n,k)*3^k. - Philippe Deléham, Mar 07 2014

Extensions

Erroneous references to spanning trees in K_2 X P_n deleted by Frans Faase, Feb 07 2009