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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, 423644570442, 1270933711326, 3812799539655, 11438398618965
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.
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
(Haskell)
a001444 n = div (3 ^ n + 3 ^ (div n 2)) 2
-- Reinhard Zumkeller, Jun 30 2013
CROSSREFS
Cf. A000244.
Sequence in context: A052870 A293743 A360274 * A293744 A293745 A293746
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
Interpretation in terms of bending wire from Colin Mallows.
STATUS
approved