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

1,1

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).

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)

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]

Prog

(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

Crossrefs

Cf. A005248, A206776.

Sequence in context: A211944 A336998 A276474 * A184438 A134135 A292279

Adjacent sequences: A362574 A362575 A362576 * A362578 A362579 A362580

Keyword

nonn,easy

Author

Eric W. Weisstein, Apr 25 2023

Extensions

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

Status

approved