A389179 - OEIS

%I #30 Feb 16 2026 09:21:02

%S 78,190,504,1562,5142,17236,58126,196432,664294,2247036,7601390,

%T 25715016,86992950,294294692,995591438,3368061312,11394069382,

%U 38545860172,130399710446,441139060088,1492362750038,5048627019764,17079382868494,57779138374320,195465425010214

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

%H Andrew Howroyd, <a href="/A389179/b389179.txt">Table of n, a(n) for n = 3..1000</a>

%H Andrew Howroyd, <a href="/A389179/a389179.txt">Visibility polynomial of the n-gear graph</a>, Jan 2026.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GearGraph.html">Gear Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VisibilityPolynomial.html">Visibility Polynomial</a>.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6,2,-1,1).

%F 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

%t 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 *)

%t Join[{78, 190, 504, 1562}, LinearRecurrence[{5, -6, 2, -1, 1}, {5142, 17236, 58126, 196432, 664294}, 20]] (* _Eric W. Weisstein_, Feb 16 2026 *)

%t 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 *)

%o (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

%K nonn,easy

%O 3,1

%A _Eric W. Weisstein_, Sep 25 2025

%E a(13) from _Eric W. Weisstein_, Oct 18 2025

%E a(14) from _Eric W. Weisstein_, Oct 23 2025

%E a(15) onward from _Andrew Howroyd_, Dec 27 2025