A290938 -id:A290938 - cp's OEIS frontend

A290378 Number of minimal dominating sets in the n-gear graph.

2, 8, 8, 16, 37, 80, 156, 304, 602, 1173, 2290, 4456, 8686, 16892, 32833, 63776, 123864, 240524, 467060, 907061, 1761894, 3423164, 6652706, 12933280, 25151787, 48931280, 95228360, 185400336, 361093444, 703546005, 1371282460, 2673742784, 5215147858

Offset: 1

Eric W. Weisstein, Jul 28 2017

nonn

Sequence extrapolated to n = 1 using recurrence. - Andrew Howroyd, Aug 27 2017

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 Dominating Set
Index entries for linear recurrences with constant coefficients, signature (4, -3, -4, 4, -1, 1, 3, -3, 0, 2, 3, 0, -1).

Cf. A290589, A290938.

Table[RootSum[-1 - # - #^2 + #^3 &, #^n &] + RootSum[1 - 2 # - #^2 + 3 #^3 - #^4 - 2 #^5 + #^6 &, #^n &] - LucasL[n] - 2 Cos[n Pi/2], {n, 20}]
LinearRecurrence[{4, -3, -4, 4, -1, 1, 3, -3, 0, 2, 3, 0, -1}, {2, 8, 8, 16, 37, 80, 156, 304, 602, 1173, 2290, 4456, 8686}, 20]
CoefficientList[Series[(2 - 18 x^2 + 16 x^3 + 21 x^4 - 18 x^5 - 15 x^6 - 2 x^7 + 16 x^8 + 2 x^9 + 11 x^10 - 2 x^11 - 5 x^12)/((1 + x^2) (1 - x - x^2) (1 - x - x^2 - x^3) (1 - 2 x - x^2 + 3 x^3 - x^4 - 2 x^5 + x^6)), {x, 0, 20}], x]

Vec((2 - 18*x^2 + 16*x^3 + 21*x^4 - 18*x^5 - 15*x^6 - 2*x^7 + 16*x^8 + 2*x^9 + 11*x^10 - 2*x^11 - 5*x^12)/((1 + x^2)*(1 - x - x^2)*(1 - x - x^2 - x^3)*(1 - 2*x - x^2 + 3*x^3 - x^4 - 2*x^5 + x^6)) + O(x^30)) \\ Andrew Howroyd, Aug 27 2017

From Andrew Howroyd, Aug 27 2017: (Start)
a(n) = 4*a(n-1) - 3*a(n-2) - 4*a(n-3) + 4*a(n-4) - a(n-5) + a(n-6) + 3*a(n-7) - 3*a(n-8) + 2*a(n-10) + 3*a(n-11) - a(n-13) for n > 13.
G.f.: x*(2 - 18*x^2 + 16*x^3 + 21*x^4 - 18*x^5 - 15*x^6 - 2*x^7 + 16*x^8 + 2*x^9 + 11*x^10 - 2*x^11 - 5*x^12)/((1 + x^2)*(1 - x - x^2)*(1 - x - x^2 - x^3)*(1 - 2*x - x^2 + 3*x^3 - x^4 - 2*x^5 + x^6)).
(End)

a(13)-a(24) from Andrew Howroyd, Aug 11 2017

a(1)-a(2) and terms a(25) and beyond from Andrew Howroyd, Aug 27 2017

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

A290378 Number of minimal dominating sets in the n-gear graph.

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