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%I #26 Dec 07 2019 12:18:22
%S 0,2,8,28,96,328,1120,3824,13056,44576,152192,519616,1774080,6057088,
%T 20680192,70606592,241065984,823050752,2810071040,9594182656,
%U 32756588544,111837988864,381838778368,1303679135744
%N Number of routes of length 2n on the sides of an octagon from a point to opposite point.
%C Also the 2nd row in the 2-shuffle Phi_2(W(sqrt(2)) of the Fraenkel-Kimberling publication. [_R. J. Mathar_, Aug 17 2009].
%C First differences of A056236. - _Jeremy Gardiner_, Aug 11 2013
%H Harry J. Smith, <a href="/A060995/b060995.txt">Table of n, a(n) for n=1,...,200</a>
%H Tomislav Doslic, I. Zubac, <a href="http://amc-journal.eu/index.php/amc/article/view/851">Counting maximal matchings in linear polymers</a>, Ars Mathematica Contemporanea 11 (2016) 255-276.
%H International Mathematical Olympiad, <a href="http://www.kalva.demon.co.uk/imo/isoln/isoln796.html">1979 Problem 6</a>
%H A. S. Fraenkel, C. Kimberling, <a href="http://dx.doi.org/10.1016/0012-365X(94)90259-3">Generalized Wythoff arrays, shuffles and interspersions</a>, Discr. Math. 126 (1-3) (1994) 137-149. [From _R. J. Mathar_, Aug 17 2009]
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4, -2).
%F G.f.: 2*x^2/(1-4*x+2*x^2).
%F a(n) = (2 + sqrt(2))^(n-1)/sqrt(2) - (2-sqrt(2))^(n-1)/sqrt(2).
%F a(n) = 4*a(n-1)-2*a(n-2).
%F a(n) = 2*A007070(n-2)
%F G.f.: G(0)/(2*x) - 1/x, where G(k)= 1 + 1/( 1 - 4*x^2/(4*x^2 + 2*(1-2*x)^2/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 16 2013
%t LinearRecurrence[{4,-2},{0,2},40] (* _Harvey P. Dale_, Mar 03 2012 *)
%o (PARI) { for (n=1, 200, if (n>2, a=4*a1 - 2*a2; a2=a1; a1=a, if (n==1, a=a2=0, a=a1=2)); write("b060995.txt", n, " ", a) ) } \\ _Harry J. Smith_, Jul 16 2009
%o (Sage) [(lucas_number2(n,4,2)-lucas_number2(n-1,4,2)) for n in range(0, 24)] # _Zerinvary Lajos_, Nov 10 2009
%K nonn
%O 1,2
%A _Henry Bottomley_, May 13 2001
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