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A003683 2^(n-1)*(2^n - (-1)^n)/3. 21
0, 1, 2, 12, 40, 176, 672, 2752, 10880, 43776, 174592, 699392, 2795520, 11186176, 44736512, 178962432, 715816960, 2863333376, 11453202432, 45813071872, 183251763200, 733008101376, 2932030308352, 11728125427712 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivré Formula, March 2014; Preprint on ResearchGate.

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

Eric Weisstein's World of Mathematics, Octahedral Graph

Index entries for linear recurrences with constant coefficients, signature (2,8).

FORMULA

a(1) = 1, a(2) = 2; a(n) = 2a(n-1) + 8a(n-2). - 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) = A001045(n+1) * A000079(n). - R. J. Mathar, Jul 08 2009

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

MAPLE

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

MATHEMATICA

a[n_]:=(MatrixPower[{{1, 5}, {1, -3}}, n].{{1}, {1}})[[2, 1]]; Table[Abs[a[n]], {n, -1, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)

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 *)

PROG

(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 xrange(0, 24)] # Zerinvary Lajos, Apr 22 2009

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

CROSSREFS

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

Sequence in context: A190064 A240122 A110953 * A188572 A098519 A127725

Adjacent sequences:  A003680 A003681 A003682 * A003684 A003685 A003686

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified November 19 08:51 EST 2017. Contains 294923 sequences.