A364605 Number of 6-cycles in the n-Lucas cube graph.

0, 0, 0, 0, 5, 44, 147, 464, 1236, 3100, 7293, 16472, 35919, 76216, 158040, 321472, 643229, 1268868, 2472147, 4764120, 9092300, 17202636, 32294277, 60199088, 111498175, 205306192, 376014960, 685273120, 1243205205, 2245893340, 4041415347, 7245914176, 12947137412

Offset: 1

Author

Eric W. Weisstein, Jul 30 2023

Keywords

• nonn
• easy

Links

• Eric Weisstein's World of Mathematics, Graph Cycle
• Eric Weisstein's World of Mathematics, Lucas Cube Graph
• Index entries for linear recurrences with constant coefficients, signature (4,-2,-8,5,8,-2,-4,-1).

Crossrefs

Cf. A245961 (number of 4-cycles).

Programs

Mathematica

Join[{0}, Table[(n + 1) (3 (40 n^2 - 145 n + 99) Fibonacci[n] - (40 n^2 - 133 n + 75) LucasL[n])/150, {n, 20}]]
Join[{0}, LinearRecurrence[{4, -2, -8, 5, 8, -2, -4, -1}, {0, 0, 0, 5, 44, 147, 464, 1236}, 20]]
CoefficientList[Series[x^4 (5 + 24 x - 19 x^2 + 4 x^3 + x^4)/(-1 + x + x^2)^4, {x, 0, 20}], x]

Formula

a(n) = (n + 1)*(3*(40n^2 - 145*n + 99)*A000045(n) - (40*n^2 - 133*n + 75)*A000032(n))/150.

a(n) = 4*a(n-1) - 2*a(n-2) - 8*a(n-3) + 5*a(n-4) + 8*a(n-5) - 2*a(n-6) - 4*a(n-7) - a(n-8) for n > 1.

G.f.: x^4*(5+24*x-19*x^2+4*x^3+x^4)/(-1+x+x^2)^4.