A389179 Number of mutual-visibility sets in the n-gear graph.

78, 190, 504, 1562, 5142, 17236, 58126, 196432, 664294, 2247036, 7601390, 25715016, 86992950, 294294692, 995591438, 3368061312, 11394069382, 38545860172, 130399710446, 441139060088, 1492362750038, 5048627019764, 17079382868494, 57779138374320, 195465425010214

Offset

3,1

Links

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

Andrew Howroyd, Visibility polynomial of the n-gear graph, Jan 2026.

Eric Weisstein's World of Mathematics, Gear Graph.

Eric Weisstein's World of Mathematics, Visibility Polynomial.

Index entries for linear recurrences with constant coefficients, signature (5,-6,2,-1,1).

Formula

G.f.: 2*x^3*(39 - 100*x + 11*x^2 + 13*x^3 + 27*x^4 + x^5 - 6*x^6 - 4*x^7 - x^8)/((1 - x)^2*(1 - 3*x - x^2 - x^3)). - Andrew Howroyd, Dec 27 2025

Mathematica

Table[Piecewise[{{78, n == 3}, {190, n == 4}, {504, n == 5}, {1562, n == 6}}, 10 n + 1 + RootSum[-1 - # - 3 #^2 + #^3 &, #^n &]], {n, 3, 20}] (* Eric W. Weisstein, Feb 16 2026 *)
Join[{78, 190, 504, 1562}, LinearRecurrence[{5, -6, 2, -1, 1}, {5142, 17236, 58126, 196432, 664294}, 20]] (* Eric W. Weisstein, Feb 16 2026 *)
CoefficientList[Series[78 + 190 x + (2 x^2 (-252 + 479 x - 178 x^2 + 55 x^3 - 89 x^4 + 4 x^5 + x^6))/((-1 + x)^2 (-1 + 3 x + x^2 + x^3)), {x, 0, 20}], x] (* Eric W. Weisstein, Feb 16 2026 *)

Prog

(PARI) Vec(2*x^3*(39 - 100*x + 11*x^2 + 13*x^3 + 27*x^4 + x^5 - 6*x^6 - 4*x^7 - x^8)/((1 - x)^2*(1 - 3*x - x^2 - x^3)) + O(x^31)) \\ Andrew Howroyd, Jan 09 2026

Crossrefs

Sequence in context: A044410 A044791 A389203 * A232386 A257484 A068130

Adjacent sequences: A389176 A389177 A389178 * A389180 A389181 A389182

Keyword

nonn,easy

Author

Eric W. Weisstein, Sep 25 2025

Extensions

a(13) from Eric W. Weisstein, Oct 18 2025

a(14) from Eric W. Weisstein, Oct 23 2025

a(15) onward from Andrew Howroyd, Dec 27 2025

Status

approved