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 A358917 #47 May 07 2024 07:32:32
%S A358917 0,1,15,80,609,4015,27936,190385,1307775,8956144,61405905,420831071,
%T A358917 2884553280,19770670945,135511114479,928804587920,6366127657281,
%U A358917 43634071586575,299072419071840,2049872742473489,14050037090947935,96300386075488816,660052667580788145
%N A358917 a(n) = Fibonacci(n+1)^4 - Fibonacci(n-1)^4.
%C A358917 a(n) is the number of edge covers of a spider graph with four branches where each branch has n vertices besides the center vertex.
%C A358917 An edge cover of a graph is a subset of edges for which each vertex is incident to at least one edge in the subset.
%C A358917 The idea is each branch is treated as a path P_(n+2). Each branch acts independently then and has F(n+1) covers (P_n has F(n-1) covers), hence F(n+1)^4 total. Except we remove the cases where each branch is missing the connecting edge to the center, which is when that edge cover comes from P_n , hence the minus F(n-1)^4.
%H A358917 Michael De Vlieger, <a href="/A358917/b358917.txt">Table of n, a(n) for n = 0..1196</a>
%H A358917 Feryal Alayont and Evan Henning, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Alayont/ala4.html">Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
%H A358917 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,19,4,-1).
%F A358917 From _Stefano Spezia_, Dec 06 2022: (Start)
%F A358917 G.f.: x*(1 + 11*x + x^2)/((1 - 7*x + x^2)*(1 + 3*x + x^2)).
%F A358917 a(n) = 4*a(n-1) + 19*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. (End)
%F A358917 5*a(n) = 9*A004187(n) + 4*(-1)^n*A001906(n) . - _R. J. Mathar_, May 07 2024
%e A358917 Case n=1 is a star graph with four branches and one edge cover (all edges).
%e A358917 * *
%e A358917 \ /
%e A358917 *__C__*
%e A358917 For n=2, there are 15 edge covers of the graph obtained by gluing four P_3 paths at one single vertex. Each of the pendant edges of the P_3's have to be in the edge cover for the pendants to be incident with an edge. The middle vertices are then automatically incident with at least one edge. There remains the center vertex. We then need at least one of the remaining four edges to be in the subset, giving us 2^4-1 choices.
%e A358917 * *
%e A358917 \ /
%e A358917 * *
%e A358917 \ /
%e A358917 *__*__C__*__*
%t A358917 LinearRecurrence[{4, 19, 4, -1}, {0, 1, 15, 80}, 25] (* _Amiram Eldar_, Dec 06 2022 *)
%o A358917 (Python)
%o A358917 from sympy import fibonacci
%o A358917 def a(n): return fibonacci(n+1)**4-fibonacci(n-1)**4
%Y A358917 Cf. A000045, A056571, A358934.
%K A358917 nonn,easy
%O A358917 0,3
%A A358917 _Feryal Alayont_, Dec 05 2022