A295420 Number of total dominating sets in the n-Moebius ladder.

1, 11, 49, 131, 441, 1499, 5041, 17155, 58081, 196331, 664225, 2246915, 7601049, 25714875, 86992929, 294294531, 995591809, 3368061131, 11394068049, 38545859971, 130399709881, 441139059867, 1492362754129, 5048627019523, 17079382863841, 57779138374059

Offset: 1

Author

Eric W. Weisstein, Apr 16 2018

Keywords

• nonn
• easy

Comments

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, Apr 16 2018

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Moebius Ladder
• Eric Weisstein's World of Mathematics, Total Dominating Set
• Index entries for linear recurrences with constant coefficients, signature (3,0,4,-2,10,4,0,-1,-1).

Crossrefs

Cf. A284663, A290337.

Programs

Mathematica

Table[RootSum[-1 - # - 3 #^2 + #^3 &, #^n &] - RootSum[1 + # - #^2 + #^3 &, #^n &] + RootSum[-1 + # + #^2 + #^3 &, #^n &], {n, 20}]
LinearRecurrence[{3, 0, 4, -2, 10, 4, 0, -1, -1}, {1, 11, 49, 131, 441, 1499, 5041, 17155, 58081}, 20]
CoefficientList[Series[(1 + 8 x + 16 x^2 - 20 x^3 + 6 x^4 - 8 x^5 + 4 x^6 - 4 x^7 - 3 x^8)/(1 - 3 x - 4 x^3 + 2 x^4 - 10 x^5 - 4 x^6 + x^8 + x^9), {x, 0, 20}], x]

PARI

Vec((1 - x)*(1 + 9*x + 25*x^2 + 5*x^3 + 11*x^4 + 3*x^5 + 7*x^6 + 3*x^7)/((1 - x + x^2 + x^3)*(1 + x + x^2 - x^3)*(1 - 3*x - x^2 - x^3)) + O(x^30)) \\ Andrew Howroyd, Apr 16 2018

Formula

From Andrew Howroyd, Apr 16 2018: (Start)

G.f.: x*(1 - x)*(1 + 9*x + 25*x^2 + 5*x^3 + 11*x^4 + 3*x^5 + 7*x^6 + 3*x^7)/((1 - x + x^2 + x^3)*(1 + x + x^2 - x^3)*(1 - 3*x - x^2 - x^3)).

a(n) = 3*a(n-1) + 4*a(n-3) - 2*a(n-4) + 10*a(n-5) + 4*a(n-6) - a(n-8) - a(n-9) for n > 9. (End)

Extensions

a(1)-a(2) and terms a(10) and beyond from Andrew Howroyd, Apr 16 2018