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
a(n) = (3^n + 3^floor(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 by 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
From Colin Barker, Apr 02 2012: (Start)
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)). (End)
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
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
Interpretation in terms of bending wire from Colin Mallows.
STATUS
approved