%I #29 Mar 22 2022 17:45:40
%S 1,2,13,52,273,1302,6573,32552,163073,813802,4070573,20345052,
%T 101733073,508626302,2543170573,12715657552,63578483073,317891438802,
%U 1589458170573,7947285970052,39736434733073,198682149251302,993410770670573,4967053731282552
%N Number of walks of length n between two vertices on an icosahedron at distance 1.
%H Colin Barker, <a href="/A030517/b030517.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,10,-20,-25).
%F a(n) = 2*a(n-1) + 2*A030518(n-1) + 5*a(n-2).
%F A030518(n) = 2*a(n-1) + 2*A030518(n-1) + 5*A030518(n-2).
%F From _Emeric Deutsch_, Apr 03 2004: (Start)
%F a(n) = 5^n/12 - (-1)^n/12 + (sqrt(5))^(n+1)/20 + (-sqrt(5))^(n+1)/20.
%F a(n) = 4*a(n-1) + 10*a(n-2) - 20*a(n-3) - 25*a(n-4) for n>=5. (End)
%F From _Colin Barker_, Oct 17 2016: (Start)
%F G.f.: x*(1 - 2*x - 5*x^2)/((1 + x)*(1 - 5*x)*(1 - 5*x^2)).
%F a(n) = (5^n - 1)/12 for n even.
%F a(n) = (6*5^((n-1)/2) + 5^n + 1)/12 for n odd. (End)
%t LinearRecurrence[{4, 10, -20, -25}, {1, 2, 13, 52}, 24] (* _Jean-François Alcover_, Jul 12 2021 *)
%o (PARI) Vec(x*(1-2*x-5*x^2)/((1+x)*(1-5*x)*(1-5*x^2)) + O(x^30)) \\ _Colin Barker_, Oct 17 2016
%Y Cf. A030518.
%K nonn,walk,easy
%O 1,2
%A _Yasutoshi Kohmoto_