A387571 Number of matchings in the n-double cone graph.

3, 21, 64, 217, 671, 2052, 6119, 18001, 52288, 150429, 429243, 1216516, 3427659, 9609357, 26821696, 74576737, 206650199, 570877956, 1572754223, 4322192329, 11851475008, 32430381861, 88576465779, 241511251972, 657457204371, 1787147867397, 4851349002304

Offset: 1

Author

Eric W. Weisstein, Sep 02 2025

Keywords

• nonn
• easy
• new

Comments

Sequence extended to a(1) using the formula/recurrence.

Links

• Eric Weisstein's World of Mathematics, Double Cone Graph.
• Eric Weisstein's World of Mathematics, Independent Edge Set.
• Index entries for linear recurrences with constant coefficients, signature (5,-5,-5,5,-1).

Programs

Mathematica

Table[2 (-1)^(n + 1) + Fibonacci[2 n + 5] + ((5 - 3 Sqrt[5]) (3 - Sqrt[5])^n + (3 + Sqrt[5])^n (5 + 3 Sqrt[5])) n/(5 2^n) , {n, 0, 20}] // Expand
seq = LinearRecurrence[{5, -5, -5, 5, -1}, {3, 21, 64, 217, 671, 2052}, 20]
CoefficientList[Series[(3 + 6 x - 26 x^2 + 17 x^3 - 4 x^4)/((1 + x) (1 - 3 x + x^2)^2), {x, 0, 20}], x]

Formula

a(n) = 5*a(n-1)-5*a(n-2)-5*a(n-3)+5*a(n-4)-a(n-5).

G.f.: (x*(3+6*x-26*x^2+17*x^3-4*x^4))/((1+x)*(1-3*x+x^2)^2).