A362577 Number of vertex cuts in the n-trapezohedral graph.

5, 15, 88, 435, 1957, 8394, 35273, 146795, 607492, 2503687, 10282873, 42103670, 171925709, 700339023, 2846710048, 11549292123, 46778169517, 189188288130, 764162167025, 3083079787091, 12426568931356, 50042249662927, 201366368701441, 809732016511598, 3254128933657397

Offset: 1

Author

Eric W. Weisstein, Apr 25 2023

Keywords

• nonn
• easy

Comments

The n-trapezohedral graph is defined for n >= 3. The sequence has been extended to n=1 using the formula/recurrence. - Andrew Howroyd, May 03 2023

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500
• Eric Weisstein's World of Mathematics, Trapezohedral Graph
• Eric Weisstein's World of Mathematics, Vertex Cut
• Index entries for linear recurrences with constant coefficients, signature (13,-65,156,-179,69,37,-38,8).

Crossrefs

Cf. A005248, A206776.

Programs

Mathematica

Table[LucasL[2 n] - ((3 - Sqrt[17])^n + (3 + Sqrt[17])^n)/2^(n - 1) + 2 n - 4 n^2 + 3 4^n - 2, {n, 20}] //Expand
LinearRecurrence[{13, -65, 156, -179, 69, 37, -38, 8}, {5, 15, 88, 435, 1957, 8394, 35273, 146795}, 20]
CoefficientList[Series[(-5 + 50 x - 218 x^2 + 514 x^3 - 577 x^4 + 160 x^5 - 28 x^6 + 8 x^7)/((-1 + x)^3 (-1 + 4 x) (1 - 3 x + x^2) (-1 + 3 x + 2 x^2)), {x, 0, 20}], x]

PARI

Vec((5 - 50*x + 218*x^2 - 514*x^3 + 577*x^4 - 160*x^5 + 28*x^6 - 8*x^7)/((1 - x)^3*(1 - 4*x)*(1 - 3*x + x^2)*(1 - 3*x - 2*x^2)) + O(x^30)) \\ Andrew Howroyd, May 03 2023

Formula

From Andrew Howroyd, May 03 2023: (Start)

a(n) = 3*4^n - 4*n^2 + 2*n - 2 + A005248(n) - 2*A206776(n).

a(n) = 13*a(n-1) - 65*a(n-2) + 156*a(n-3) - 179*a(n-4) + 69*a(n-5) + 37*a(n-6) - 38*a(n-7) + 8*a(n-8) for n > 8.

G.f.: x*(5 - 50*x + 218*x^2 - 514*x^3 + 577*x^4 - 160*x^5 + 28*x^6 - 8*x^7)/((1 - x)^3*(1 - 4*x)*(1 - 3*x + x^2)*(1 - 3*x - 2*x^2)).

(End)

Extensions

a(1)-a(2) prepended and a(15) and beyond from Andrew Howroyd, May 03 2023