This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380461 #23 Jun 24 2025 13:30:38
%S A380461 1,16,154,1289,10180,78372,596337,4512900,34064998,256825009,
%T A380461 1935169456,14577526976,109797758833,826945679592,6227993359362,
%U A380461 46904386459065,353244994467916,2660340755025580,20035394638446769,150889230111278492,1136366561949728110
%N A380461 Number of edge covers of fan graph F_{2,n}.
%C A380461 The fan graph F_{2,n} is the join of the path graph P_n with two isolated vertices. That is, two new vertices are added and each is connected to every vertex in P_n. a(n) is the number of edge covers of F_{2,n}.
%H A380461 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FanGraph.html">Fan Graph</a>
%H A380461 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (11,-24,-21,33,34,8).
%F A380461 a(n) = 11*a(n-1) - 24*a(n-2) - 21*a(n-3) + 33*a(n-4) + 34*a(n-5) + 8*a(n-6), for n >= 7.
%e A380461 For n = 1, the fan graph F_{2,1} becomes a path with three vertices. There is exactly 1 edge cover.
%e A380461 For n = 2, the base graph P_2 has endpoints v and w, and two added vertices a and b each connect to both v and w. Each of a and b has 3 edge-covering options: use the edge to v, the edge to w, or both. This gives 3 × 3 = 9 combinations. For each, the edge (v,w) from P_2 may be included or not, giving 18 configurations. However, in 2 of these, omitting the edge (v,w) leaves v or w uncovered. Therefore, the number of valid edge covers is 18 - 2 = 16.
%K A380461 nonn,easy
%O A380461 1,2
%A A380461 _Feryal Alayont_, Jun 22 2025