%I #15 Feb 12 2022 15:30:02
%S 1,2,5,12,30,76,196,512,1352,3600,9648,25984,70240,190400,517184,
%T 1406976,3831936,10445056,28488448,77735936,212186624,579320832,
%U 1581966336,4320477184,11800692736,32233951232
%N Expansion of (1-2*x-x^2)/((1-2*x)*(1-2*x+2*x^2)).
%C Number of (s(0), s(1), ..., s(n+1)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n+1, s(0) = 1, s(n+1) = 2. - _Herbert Kociemba_, Jun 17 2004
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-4).
%F a(n) = (3*2^(n+1) - sqrt(3)*(1-sqrt(3))^(n+1) + sqrt(3)*(1+sqrt(3))^(n+1))/12. - _Herbert Kociemba_, Jun 17 2004
%F 2*a(n) = 2^n + A002605(n+1). - _R. J. Mathar_, Sep 11 2019
%F a(n) = 4*a(n-1)-2*a(n-2)-4*a(n-3). - _Wesley Ivan Hurt_, May 14 2021
%o (PARI) Vec((1-2*x-x^2)/(1-4*x+2*x^2+4*x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%Y First differences are in A094297.
%Y Cf. A002605.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_