A296102 Number of total dominating sets in the n-prism graph.

3, 9, 39, 121, 443, 1521, 5071, 17161, 58035, 196249, 664183, 2247001, 7601259, 25715041, 86992799, 294294025, 995591267, 3368061225, 11394069191, 38545861561, 130399710235, 441139057489, 1492362749807, 5048627017225, 17079382868243, 57779138385081, 195465425009943

Offset

1,1

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, Prism Graph

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).

Formula

From Andrew Howroyd, Apr 16 2018: (Start)

G.f.: x*(3 + 12*x^2 - 8*x^3 + 50*x^4 + 24*x^5 - 8*x^7 - 9*x^8)/((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)

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}, {3, 9, 39, 121, 443,
   1521, 5071, 17161, 58035}, 20]
CoefficientList[Series[(3 + 12 x^2 - 8 x^3 + 50 x^4 + 24 x^5 - 8 x^7 - 9 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]

Prog

(PARI) Vec((3 + 12*x^2 - 8*x^3 + 50*x^4 + 24*x^5 - 8*x^7 - 9*x^8)/((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

Crossrefs

Cf. A284702, A290336, A295420.

Sequence in context: A270593 A059804 A065657 * A149026 A149027 A330795

Adjacent sequences: A296099 A296100 A296101 * A296103 A296104 A296105

Keyword

nonn,easy

Author

Eric W. Weisstein, Apr 16 2018

Extensions

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

Status

approved