A290589 Number of irredundant sets in the n-gear graph.

41, 113, 320, 908, 2574, 7317, 20855, 59582, 170681, 490180, 1411207, 4072092, 11774998, 34114389, 99006951, 287783255, 837649471, 2441108294, 7121588680, 20795782761, 60775937523, 177746557324, 520168686310, 1523090681155, 4461852378404, 13076323061624

Offset: 3

Author

Eric W. Weisstein, Aug 07 2017

Keywords

• nonn
• changed

Links

• Christian Sievers, Table of n, a(n) for n = 3..200
• Eric Weisstein's World of Mathematics, Gear Graph.
• Eric Weisstein's World of Mathematics, Irredundant Set.
• Index entries for linear recurrences with constant coefficients, signature (11,-45,74,1,-140,100,58,-49,19,-57,-4,40,-2,-7,-11,0,4).

Crossrefs

Cf. A290378, A290640.

Programs

Mathematica

Join[{41}, Table[2^(n - 4) n + LucasL[n] + RootSum[-1 - # - 2 #^2 + #^3 &, #^n &] - RootSum[-1 - # - #^2 - #^3 + #^4 &, #^n &] + RootSum[1 - 2 # - #^2 + 4 #^3 - #^4 - 3 #^5 + #^6 &, #^n &], {n, 4, 10}]] (* Eric W. Weisstein, Sep 04 2025 *)
Join[{41}, LinearRecurrence[{11, -45, 74, 1, -140, 100, 58, -49, 19, -57, -4, 40, -2, -7, -11, 0, 4}, {113, 320, 908, 2574, 7317, 20855, 59582, 170681, 490180, 1411207, 4072092, 11774998, 34114389, 99006951, 287783255, 837649471, 2441108294}, 20]] (* Eric W. Weisstein, Sep 04 2025 *)
CoefficientList[Series[(-41 + 338 x - 922 x^2 + 561 x^3 + 1417 x^4 - 1810 x^5 - 406 x^6 + 820 x^7 - 297 x^8 + 950 x^9 - 135 x^10 - 664 x^11 + 180 x^12 + 172 x^13 + 154 x^14 - 37 x^15 - 70 x^16 + 6 x^17)/((-1 + 2 x)^2 (-1 + x + x^2) (-1 + 2 x + x^2 + x^3) (-1 + x + x^2 + x^3 + x^4) (1 - 3 x - x^2 + 4 x^3 - x^4 - 2 x^5 + x^6)), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 04 2025 *)

Formula

a(n) = 11*a(n-1)-45*a(n-2)+74*a(n-3)+a(n-4)-140*a(n-5)+100*a(n-6)+58*a(n-7)-49*a(n-8)+19*a(n-9)-57*a(n-10)-4*a(n-11)+40*a(n-12)-2*a(n-13)-7*a(n-14)-11*a(n-15)+4*a(n-17) for n > 20. - Eric W. Weisstein, Sep 04 2025

G.f.: x^3*(-41+338*x-922*x^2+561*x^3+1417*x^4-1810*x^5-406*x^6+820*x^7-297*x^8+950*x^9-135*x^10-664*x^11+180*x^12+172*x^13+154*x^14-37*x^15-70*x^16+6*x^17)/((-1+2*x)^2*(-1+x+x^2)*(-1+2*x+x^2+x^3)*(-1+x+x^2+x^3+x^4)*(1-3*x-x^2+4*x^3-x^4-2*x^5+x^6)). - Eric W. Weisstein, Sep 04 2025

Extensions

a(13)-a(16) from Andrew Howroyd, Aug 11 2017

Missing a(12) inserted and a(17)-a(18) added by Andrew Howroyd, Aug 27 2017

More terms from Christian Sievers, Nov 18 2023