A362516 Number of vertex cuts in the n-gear graph.

1, 5, 51, 293, 1383, 6017, 25315, 104941, 431775, 1768377, 7218555, 29388325, 119381239, 484031537, 1959295251, 7919693789, 31972642767, 128937189161, 519476334379, 2091181293589, 8412008183079, 33816433653921, 135865503379395, 545598121631437, 2190000348372223

Offset: 1

Author

Eric W. Weisstein, Apr 23 2023

Keywords

• nonn
• easy

Comments

Extended to n = 1 using formula/recurrence.

Links

• Eric Weisstein's World of Mathematics, Gear Graph
• Eric Weisstein's World of Mathematics, Vertex Cut
• Index entries for linear recurrences with constant coefficients, signature (10,-34,44,-13,-14,8).

Crossrefs

Cf. A286188.

Programs

Mathematica

Table[2 (4^n - 1) + 2 n - 4 n^2 - (1/2 (3 - Sqrt[17]))^n - (1/2 (3 + Sqrt[17]))^n, {n, 20}] // Expand
LinearRecurrence[{10, -34, 44, -13, -14, 8}, {1, 5, 51, 293, 1383, 6017}, 20]
CoefficientList[Series[(-1 + 5 x - 35 x^2 + 91 x^3 + 20 x^4 + 16 x^5)/((-1 + x)^3 (1 - 7 x + 10 x^2 + 8 x^3)), {x, 0, 20}], x]

Formula

a(n) = 2^(2*n+1) - 1 - A286188(n). - Pontus von Brömssen, Apr 23 2023

a(n) = 2*(4^n - 1) + 2*n - 4*n^2 - ((3 - sqrt(17))/2)^n - ((3 + sqrt(17))/2)^n.

a(n) = 10*a(n-1)-34*a(n-2)+44*a(n-3)-13*a(n-4)-14*a(n-5)+8*a(n-6).

G.f.: x*(-1 + 5*x - 35*x^2 + 91*x^3 + 20*x^4 + 16*x^5)/((-1 + x)^3*(1 - 7*x + 10*x^2 + 8*x^3)).

a(n) = -A206776(n)+2*4^n-2-4*n^2+2*n. - R. J. Mathar, Feb 18 2024

Extensions

More terms (based on data in A286188) from Pontus von Brömssen, Apr 23 2023