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A003683
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a(n) = 2^(n-1)*(2^n - (-1)^n)/3.
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26
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0, 1, 2, 12, 40, 176, 672, 2752, 10880, 43776, 174592, 699392, 2795520, 11186176, 44736512, 178962432, 715816960, 2863333376, 11453202432, 45813071872, 183251763200, 733008101376, 2932030308352, 11728125427712
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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M. Gardner, Riddles of the Sphinx, New Mathematical Library, M.A.A., 1987, p. 145. Math. Rev. 89i:00015.
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LINKS
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FORMULA
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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
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MAPLE
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n)=if(n<0, 0, 2^(n-1)*(2^n-(-1)^n)/3)
(Sage) [lucas_number1(n, 2, -8) for n in range(0, 24)] # Zerinvary Lajos, Apr 22 2009
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Erroneous references to spanning trees in K_2 X P_n deleted by Frans Faase, Feb 07 2009
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STATUS
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approved
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