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 A001444 Bending a piece of wire of length n+1 (configurations that can only be brought into coincidence by turning the figure over are counted as different). 6
 1, 2, 6, 15, 45, 126, 378, 1107, 3321, 9882, 29646, 88695, 266085, 797526, 2392578, 7175547, 21526641, 64573362, 193720086, 581140575, 1743421725, 5230206126, 15690618378, 47071677987, 141215033961 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The wire stays in the plane, there are n bends, each is R,L or O. REFERENCES Todd Andrew Simpson, ``Combinatorial Proofs and Generalizations of Weyl's Denominator Formula,'' Ph. D. Dissertation, Penn State University, 1994. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (3, 3, -9). FORMULA (3^n + 3^[ n/2 ] )/2. G.f.: G(0) where G(k) = 1 + x*(3*3^k + 1)*(1 + 3*x*G(k+1))/(1 + 3^k) ; - Sergei N. Gladkovskii, Dec 13 2011. [Edited Michael Somos, Sep 09 2013] E.g.f. E(x)=(exp(3*x)+cosh(x*sqrt(3))+sinh(x*sqrt(3))/sqrt(3))/2=G(0) ; G(k) = 1 + x*(3*3^k+1)/((2*k+1)*(1+3^k) - 3*x*(2*k+1)*(1+3^k)/(3*x + (2*k+2)/G(k+1))) ; (continued fraction); - Sergei N. Gladkovskii, Dec 13 2011. a(n) = 3*a(n-1)+3*a(n-2)-9*a(n-3). G.f.: x*(1-x-3*x^2)/((1-3*x)*(1-3*x^2)). [Colin Barker, Apr 02 2012] EXAMPLE There are 2 ways to bend a piece of wire of length 2 (bend it or not). G.f. = 1 + 2*x + 6*x^2 + 15*x^3 + 45*x^4 + 126*x^5 + 378*x^6 + ... MAPLE f := n->(3^floor(n/2)+3^n)/2; MATHEMATICA CoefficientList[Series[(1-x-3*x^2)/((1-3*x)*(1-3*x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 15 2012 *) LinearRecurrence[{3, 3, -9}, {1, 2, 6}, 40] (* Harvey P. Dale, Dec 30 2012 *) PROG (Haskell) a001444 n = div (3 ^ n + 3 ^ (div n 2)) 2 -- Reinhard Zumkeller, Jun 30 2013 CROSSREFS Cf. A001997, A001998. Cf. A000244. Sequence in context: A264746 A052870 A293743 * A293744 A293745 A293746 Adjacent sequences:  A001441 A001442 A001443 * A001445 A001446 A001447 KEYWORD nonn,nice,easy AUTHOR todo(AT)tasimpson.com (Todd Andrew Simpson) EXTENSIONS Interpretation in terms of bending wire from Colin Mallows. STATUS approved

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Last modified May 9 22:13 EDT 2021. Contains 343746 sequences. (Running on oeis4.)