A363476 a(n) = Fibonacci(n)^2 * Fibonacci(n+1)^3.

0, 1, 8, 108, 1125, 12800, 140608, 1565109, 17333064, 192329500, 2132531225, 23651979264, 262296652032, 2908947562937, 32260582549000, 357775937196300, 3967793428038237, 44003514081895936, 488006404120114496, 5412074146674562125, 60020821224245910600

Offset: 0

Author

Feryal Alayont, Jun 03 2023

Keywords

• nonn
• easy

Comments

For n>1, a(n) is the number of edge covers of a caterpillar graph with spine P_(5n-4), one pendant attached at vertex n counting from the left end of the spine, a second one at 2n-1, a third at 3n-2 and a fourth at 4n-3. The caterpillar graph for n=3 is as follows:

* * * *

| | | |

*--*--*--v1--*--v2--*--v3--*--*--*

Each pendant edge must be included in an edge cover and hence allows the left and right sides of a vertex adjacent to a pendant to be independent. Therefore, the caterpillar can be split into independent paths, two P_4's (ends) and three P_5's (middle). Each P_n has F_{n-1} edge covers, resulting in the a(n) expression.

The sequence also counts number of subsets of {1, 2, ..., 5n-7} which do not contain two numbers whose difference is 5, a special instance of a general result given in Math. Mag. Problem 1854 (see Links). The equivalence to the above description can be seen as follows. Every vertex except v1, v2 and v3 is incident with at least one of the pendant edges. Therefore, if we label the middle eight edges in the spine with numbers 4, 1, 6, 2, 7, 3, 8, 5 (starting from the left), the edges have to be chosen so that both 1,6, both 2,7, and both 3,8 cannot be missing. This corresponds to choosing subsets of {1, 2, ..., 8} which do not contain two numbers whose difference is 5.

References

• F. Alayont and E. Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs; submitted.

Links

• José Heber Nieto, Problem 1854 solution, Mathematics Magazine, Vol. 84, No.4 (2011), pp. 300-301.
• Marian Tetiva, Problem 1854, Mathematics Magazine, Vol. 83, No.4 (2010), p. 304.
• Index entries for linear recurrences with constant coefficients, signature (8,40,-60,-40,8,1).

Crossrefs

Cf. A000045, A066258, A197424.

Programs

Mathematica

a[n_] := Fibonacci[n]^2 * Fibonacci[n+1]^3; Array[a, 21, 0] (* Amiram Eldar, Jun 06 2023 *)

Formula

G.f.: x*(1+4*x^2+x^3) / ((1-x-x^2)*(1-11*x-x^2)*(1+4*x-x^2)).