A387570 Number of matchings in the n-dipyramidal graph.

2, 3, 9, 22, 51, 111, 234, 477, 951, 1858, 3573, 6777, 12706, 23583, 43389, 79214, 143631, 258843, 463914, 827337, 1468827, 2597018, 4574529, 8030037, 14050946, 24514011, 42651729, 74021062, 128157771, 221397543, 381680298, 656721237, 1127890911, 1933765618

Offset: 0

Author

Eric W. Weisstein, Sep 02 2025

Keywords

• nonn
• easy
• new

Comments

Sequence extended to n = 0 using formula/recurrence.

Links

• Eric Weisstein's World of Mathematics, Dipyramidal Graph.
• Eric Weisstein's World of Mathematics, Independent Edge Set.
• Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Programs

Mathematica

Table[((n^2 + 5) LucasL[n] + 9 n Fibonacci[n])/5, {n, 0, 20}]
LinearRecurrence[{3, 0, -5, 0, 3, 1}, {3, 9, 22, 51, 111, 234}, {0, 20}]
CoefficientList[Series[(-2 + 3 x - 5 x^3 + 3 x^5)/(-1 + x + x^2)^3, {x, 0, 20}], x]

Formula

a(n) = (n^2 + 5)*LucasL(n) + 9*n*Fibonacci(n))/5.

a(n) = 3*a(n-1)-5*a(n-3)+3*a(n-5)+a(n-6).

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