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Number of walks of length n between two vertices on an icosahedron at distance 2.
4

%I #22 Mar 22 2022 18:24:39

%S 0,2,8,52,248,1302,6448,32552,162448,813802,4067448,20345052,

%T 101717448,508626302,2543092448,12715657552,63578092448,317891438802,

%U 1589456217448,7947285970052,39736424967448,198682149251302,993410721842448,4967053731282552

%N Number of walks of length n between two vertices on an icosahedron at distance 2.

%H Colin Barker, <a href="/A030518/b030518.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*A030517(n-1) + 2*a(n-1) + 5*a(n-2).

%F A030517(n) = 2*A030517(n-1) + 2*a(n-1) + 5*A030517(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). (End)

%F From _Colin Barker_, Oct 17 2016: (Start)

%F G.f.: 2*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)

%o (PARI) concat(0, Vec(2*x^2/((1+x)*(1-5*x)*(1-5*x^2)) + O(x^30))) \\ _Colin Barker_, Oct 17 2016

%Y Cf. A030517.

%K nonn,walk,easy

%O 1,2

%A _Yasutoshi Kohmoto_