A389234 Number of mutual-visibility sets in the n-double cone graph.

79, 285, 1065, 4137, 16427, 65585, 262199, 1048637, 4194371, 16777289, 67108943, 268435541, 1073741915, 4294967393, 17179869287, 68719476845, 274877907059, 1099511627897, 4398046511231, 17592186044549, 70368744177803, 281474976710801, 1125899906842775

Offset

3,1

Comments

For n >= 7, the visibility polynomial is (1 + x)^(2*n) + x + 2*n*x^2*(1 + 2*x). - Andrew Howroyd, Dec 27 2025

Links

Andrew Howroyd, Table of n, a(n) for n = 3..1000

Andrew Howroyd, Visibility polynomial of cone graphs, Jan 2026.

Eric Weisstein's World of Mathematics, Double Cone Graph.

Eric Weisstein's World of Mathematics, Visibility Polynomial.

Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Formula

From Andrew Howroyd, Dec 27 2025: (Start)

a(n) = 4^n + 6*n + 1 for n >= 7.

G.f.: x^3*(79 - 189*x + 66*x^2 - 4*x^3 + 50*x^4 - 4*x^5 - 16*x^6)/((1 - x)^2*(1 - 4*x)). (End)

Mathematica

Table[Piecewise[{{79, n == 3}, {285, n == 4}, {1065, n == 5}, {4137, n == 6}}, 1 + 4^n + 6 n], {n, 3, 20}] (* Eric W. Weisstein, Feb 16 2026 *)
Join[{79, 285, 1065, 4137}, LinearRecurrence[{6, -9, 4}, {16427, 65585, 262199}, 20]] (* Eric W. Weisstein, Feb 16 2026 *)
CoefficientList[Series[(-79 + 189 x - 66 x^2 + 4 x^3 - 50 x^4 + 4 x^5 + 16 x^6)/((-1 + x)^2 (-1 + 4 x)), {x, 0, 20}], x] (* Eric W. Weisstein, Feb 16 2026 *)

Prog

(PARI) a(n) = if(n<7, [79, 285, 1065, 4137][n-2], 4^n + 6*n + 1) \\ Andrew Howroyd, Dec 27 2025

Crossrefs

Cf. A194455.

Sequence in context: A180455 A219478 A082077 * A341182 A158769 A158774

Adjacent sequences: A389231 A389232 A389233 * A389235 A389236 A389237

Keyword

nonn,easy

Author

Eric W. Weisstein, Sep 26 2025

Extensions

a(12) onward from Andrew Howroyd, Dec 27 2025

Status

approved