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%I #51 Mar 25 2024 17:13:35
%S 1,2,6,15,45,126,378,1107,3321,9882,29646,88695,266085,797526,2392578,
%T 7175547,21526641,64573362,193720086,581140575,1743421725,5230206126,
%U 15690618378,47071677987,141215033961,423644570442,1270933711326,3812799539655,11438398618965
%N 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).
%C The wire stays in the plane, there are n bends, each is R,L or O.
%D Todd Andrew Simpson, "Combinatorial Proofs and Generalizations of Weyl's Denominator Formula", Ph. D. Dissertation, Penn State University, 1994.
%H Vincenzo Librandi, <a href="/A001444/b001444.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Fo#fold">Index entries for sequences obtained by enumerating foldings</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,-9).
%F a(n) = (3^n + 3^floor(n/2))/2.
%F 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]
%F 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
%F From _Colin Barker_, Apr 02 2012: (Start)
%F a(n) = 3*a(n-1) + 3*a(n-2) - 9*a(n-3).
%F G.f.: x*(1-x-3*x^2)/((1-3*x)*(1-3*x^2)). (End)
%e There are 2 ways to bend a piece of wire of length 2 (bend it or not).
%e G.f. = 1 + 2*x + 6*x^2 + 15*x^3 + 45*x^4 + 126*x^5 + 378*x^6 + ...
%p f := n->(3^floor(n/2)+3^n)/2;
%t CoefficientList[Series[(1-x-3*x^2)/((1-3*x)*(1-3*x^2)),{x,0,30}],x] (* _Vincenzo Librandi_, Apr 15 2012 *)
%t LinearRecurrence[{3,3,-9},{1,2,6},40] (* _Harvey P. Dale_, Dec 30 2012 *)
%o (Haskell)
%o a001444 n = div (3 ^ n + 3 ^ (div n 2)) 2
%o -- _Reinhard Zumkeller_, Jun 30 2013
%Y Cf. A001997, A001998.
%Y Cf. A000244.
%K nonn,nice,easy
%O 0,2
%A _Todd Simpson_
%E Interpretation in terms of bending wire from _Colin Mallows_.
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