A302760 Number of total dominating sets in the n-antiprism graph.

3, 11, 54, 179, 648, 2414, 8809, 32195, 117945, 431696, 1579955, 5783294, 21168592, 77482521, 283608249, 1038086883, 3799689944, 13907938601, 50906985592, 186333942984, 682034858839, 2496440225499, 9137676323347, 33446476209566, 122423549667123

Offset: 1

Author

Eric W. Weisstein, Apr 12 2018

Keywords

• nonn
• easy

Comments

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, Apr 14 2018

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Antiprism Graph
• Eric Weisstein's World of Mathematics, Total Dominating Set
• Index entries for linear recurrences with constant coefficients, signature (3,1,6,-3,0,0,1).

Crossrefs

Cf. A302255, A302652, A302763.

Programs

Magma

I:=[3,11,54,179,648,2414,8809]; [n le 7 select I[n] else 3*Self(n-1)+Self(n-2)+6*Self(n-3)-3*Self(n-4)+Self(n-7): n in [1..30]]; // Vincenzo Librandi, Apr 15 2018

Mathematica

CoefficientList[Series[(3 + 2 x + 18 x^2 - 12 x^3 + 7 x^6)/(1 - 3 x - x^2 - 6 x^3 + 3 x^4 - x^7), {x, 0, 24}], x] (* Michael De Vlieger, Apr 14 2018 *)
LinearRecurrence[{3, 1, 6, -3, 0, 0, 1}, {3, 11, 54, 179, 648, 2414, 8809}, 20] (* Vincenzo Librandi, Apr 15 2018 *)
Table[RootSum[-1 + 3 #^3 - 6 #^4 - #^5 - 3 #^6 + #^7 &, #^n &], {n, 30}] (* Eric W. Weisstein, Apr 16 2018 *)
RootSum[-1 + 3 #^3 - 6 #^4 - #^5 - 3 #^6 + #^7 &, #^Range[30] &] (* Eric W. Weisstein, Apr 16 2018 *)

PARI

Vec((3 + 2*x + 18*x^2 - 12*x^3 + 7*x^6)/(1 - 3*x - x^2 - 6*x^3 + 3*x^4 - x^7) + O(x^25)) \\ Andrew Howroyd, Apr 14 2018

Formula

From Andrew Howroyd, Apr 14 2018: (Start)

a(n) = 3*a(n-1) + a(n-2) + 6*a(n-3) - 3*a(n-4) + a(n-7) for n > 7.

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

(End)

Extensions

a(1)-a(2) and terms a(11) and beyond from Andrew Howroyd, Apr 14 2018