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A358934 a(n) = Fibonacci(n+1)^5 - Fibonacci(n-1)^5.

%I A358934 #29 Aug 05 2024 15:21:52
%S A358934 0,1,31,242,3093,32525,368168,4051333,45064131,499200274,5538624025,
%T A358934 61414079849,681135796944,7553728681433,83772910243607,
%U A358934 929052526388050,10303364319347757,114266002348885717,1267229634537217144,14053790947047408701,155858934437282250075
%N A358934 a(n) = Fibonacci(n+1)^5 - Fibonacci(n-1)^5.
%C A358934 a(n) is the number of edge covers of a spider graph with five branches where each branch has n vertices besides the center vertex. 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)^5 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)^5.
%C A358934 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.
%H A358934 Michael De Vlieger, <a href="/A358934/b358934.txt">Table of n, a(n) for n = 0..957</a>
%H A358934 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 A358934 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (8,40,-60,-40,8,1).
%F A358934 From _Stefano Spezia_, Dec 07 2022: (Start)
%F A358934 G.f.: x*(1 + 23*x - 46*x^2 - 23*x^3 + x^4)/((1 + 4*x - x^2)*(1 - x - x^2)*(1 - 11*x - x^2)).
%F A358934 a(n) = 8*a(n-1) + 40*a(n-2) - 60*a(n-3) - 40*a(n-4) + 8*a(n-5) + a(n-6) for n > 5. (End)
%F A358934 5*a(n) = 2*A000045(n)+11*A049666(n)+8*(-1)^n*A001076(n). - _R. J. Mathar_, May 07 2024
%e A358934 Case n=1 is a star graph with five branches and one edge cover (all edges).
%e A358934    *   *
%e A358934     \ /
%e A358934   *__C__*
%e A358934      |
%e A358934      *
%e A358934 For n=2, there are 31 edge covers of the graph obtained by gluing five 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 five edges to be in the subset, giving us 2^5-1 choices.
%e A358934    *__ *   *__*
%e A358934        \ /
%e A358934   *__*__C__*__*
%e A358934         |
%e A358934         *__*
%t A358934 LinearRecurrence[{8, 40, -60, -40, 8, 1}, {0, 1, 31, 242, 3093, 32525}, 20] (* _Amiram Eldar_, Dec 07 2022 *)
%t A358934 Join[{0},#[[3]]-#[[1]]&/@Partition[Fibonacci[Range[0,30]]^5,3,1]] (* _Harvey P. Dale_, Aug 05 2024 *)
%o A358934 (Python)
%o A358934 from sympy import fibonacci
%o A358934 def a(n):
%o A358934     return fibonacci(n+1)**5-fibonacci(n-1)**5
%o A358934 (Python)
%o A358934 from gmpy2 import fib2
%o A358934 def A358934(n): return sum(f:=fib2(n))**5-f[1]**5 # _Chai Wah Wu_, Jan 04 2023
%Y A358934 Cf. A000045, A056572.
%K A358934 nonn,easy
%O A358934 0,3
%A A358934 _Feryal Alayont_, Dec 06 2022