A290378 -id:A290378 - cp's OEIS frontend

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

Eric W. Weisstein, Aug 07 2017

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).

Cf. A290378, A290640.

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 *)

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

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

A290938 Number of dominating sets in the n-gear graph.

Original entry on oeis.org

5, 23, 83, 291, 1015, 3539, 12339, 43043, 150239, 524723, 1833771, 6412467, 22437095, 78553491, 275180323, 964534339, 3382685743, 11869824179, 41673547291, 146387820371, 514484547639, 1809077492883, 6364347723667, 22400458807139, 78878848178815, 277881197881011

Offset: 1

Eric W. Weisstein, Aug 14 2017

nonn

Extended to a(1)-a(2) using the formula/recurrence.

James East, Jitender Kumar, James D.Mitchell, Wilf A. Wilson, Maximal subsemigroups of finite transformation and diagram monoids, Journal of Algebra (2018), 504, 176-216, arXiv:1706.04967 [math.GR], 2017-2018.
Eric Weisstein's World of Mathematics, Dominating Set
Eric Weisstein's World of Mathematics, Gear Graph
Index entries for linear recurrences with constant coefficients, signature (7, -12, -2, 3, 3, 2).

Cf. A290378 (minimal dominating sets).

Table[(1/2 (3 - Sqrt[17]))^n + (1/2 (3 + Sqrt[17]))^n - 1 + RootSum[-1 - # - 3 #^2 + #^3 &, #^n &], {n, 20}] // Expand
LinearRecurrence[{7, -12, -2, 3, 3, 2}, {5, 23, 83, 291, 1015, 3539}, 20]
CoefficientList[Series[(-5 + 12 x + 18 x^2 + 4 x^3 - 5 x^4 - 8 x^5)/(-1 + 7 x - 12 x^2 - 2 x^3 + 3 x^4 + 3 x^5 + 2 x^6), {x, 0, 20}], x]

a(n) = 7*a(n-1) - 12*a(n-2) - 2*a(n-3) + 3*a(n-4) + 3*a(n-5) + 2*a(n-6).

G.f.: (x (-5 + 12 x + 18 x^2 + 4 x^3 - 5 x^4 - 8 x^5))/(-1 + 7 x - 12 x^2 - 2 x^3 + 3 x^4 + 3 x^5 + 2 x^6).

A303226 Number of minimal total dominating sets in the n-gear graph.

Original entry on oeis.org

0, 6, 12, 6, 30, 30, 56, 110, 156, 306, 506, 870, 1560, 2652, 4692, 8190, 14280, 25122, 43890, 77006, 135056, 236682, 415380, 728462, 1278030, 2242506, 3934272, 6903756, 12113880, 21256710, 37301556, 65456190, 114864806, 201569006, 353722056, 620732310

Offset: 1

Eric W. Weisstein, Apr 20 2018

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, Apr 20 2018

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Gear Graph.
Eric Weisstein's World of Mathematics, Minimal Total Dominating Set.
Index entries for linear recurrences with constant coefficients, signature (1,2,1,-3,-1,-1,0,0,1).

Cf. A141310, A290378, A290938, A303834.

Table[RootSum[-1 - # + #^3 &, #^n &] + RootSum[-1 + # - 2 #^2 + #^3 &, #^n &] + 2 RootSum[-1 + #^2 + #^3 &, #^(n + 2) (1 + #) &], {n, 20}]
LinearRecurrence[{1, 2, 1, -3, -1, -1, 0, 0, 1}, {0, 6, 12, 6, 30, 30, 56, 110, 156}, 20]
CoefficientList[Series[-2 x (3 + 3 x - 9 x^2 - 3 x^3 - 3 x^4 + x^5 + 6 x^7)/(-1 + x + 2 x^2 + x^3 - 3 x^4 - x^5 - x^6 + x^9), {x, 0, 20}], x]

concat([0], Vec(2*(3 + 3*x - 9*x^2 - 3*x^3 - 3*x^4 + x^5 + 6*x^7)/((1 - 2*x + x^2 - x^3)*(1 + x - x^3)*(1 - x^2 - x^3)) + O(x^40))) \\ Andrew Howroyd, Apr 20 2018

From Andrew Howroyd, Apr 20 2018: (Start)
a(n) = a(n-1) + 2*a(n-2) + a(n-3) - 3*a(n-4) - a(n-5) - a(n-6) + a(n-9) for n > 9.
G.f.: 2*x^2*(3 + 3*x - 9*x^2 - 3*x^3 - 3*x^4 + x^5 + 6*x^7)/((1 - 2*x + x^2 - x^3)*(1 + x - x^3)*(1 - x^2 - x^3)).
(End)

a(1)-a(2) and terms a(11) and beyond from Andrew Howroyd, Apr 20 2018

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A290938 Number of dominating sets in the n-gear graph.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A303226 Number of minimal total dominating sets in the n-gear graph.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions