%I #10 Feb 16 2025 08:33:51
%S 5,33,117,322,745,1515,2793,4772,7677,11765,17325,24678,34177,46207,
%T 61185,79560,101813,128457,160037,197130,240345,290323,347737,413292,
%U 487725,571805,666333,772142,890097,1021095,1166065,1325968,1501797,1694577,1905365,2135250
%N Number of (undirected) paths in the n-gear graph.
%C Extended to a(1)-a(2) using the formula/recurrence.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GearGraph.html">Gear Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphPath.html">Graph Path</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = n*(-21 + 58*n - 15*n^2 + 8*n^3)/6.
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F G.f.: x*(-5 - 8*x - 2*x^2 - 17*x^3)/(-1 + x)^5.
%t Table[n (-21 + 58 n - 15 n^2 + 8 n^3)/6, {n, 20}]
%t LinearRecurrence[{5, -10, 10, -5, 1}, {5, 33, 117, 322, 745}, 20]
%t CoefficientList[Series[(-5 - 8 x - 2 x^2 - 17 x^3)/(-1 + x)^5, {x, 0, 20}], x]
%K nonn,easy,changed
%O 1,1
%A _Eric W. Weisstein_, Sep 07 2017