A284702 Number of dominating sets in the n-prism graph.

3, 11, 51, 183, 663, 2435, 8935, 32775, 120219, 440971, 1617531, 5933271, 21763823, 79831875, 292831311, 1074134535, 3940032883, 14452434635, 53012975555, 194456895863, 713287340551, 2616409296963, 9597250953527, 35203676264199, 129130605057163

Offset: 1

Author

Eric W. Weisstein, Apr 01 2017

Keywords

• nonn
• easy

Comments

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, May 10 2017

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000 (terms 1..200 from Andrew Howroyd)
• Eric Weisstein's World of Mathematics, Dominating Set
• Eric Weisstein's World of Mathematics, Prism Graph
• Index entries for linear recurrences with constant coefficients, signature (3,1,5,1,1,-1,-1).

Crossrefs

Cf. A051927, A102080, A284699, A218348 (ladder), A284663.

Programs

Mathematica

LinearRecurrence[{3, 1, 5, 1, 1, -1, -1}, {3, 11, 51, 183, 663, 2435,
  8935}, 20] (* Eric W. Weisstein, May 17 2017 *)
Rest[CoefficientList[Series[x (-7 x^6 - 6 x^5 + 5 x^4 + 4 x^3 + 15 x^2 + 2 x + 3)/((x^2 + 1) (x^5 + x^4 - 2 x^3 - 2 x^2 - 3 x + 1)), {x, 0, 20}], x]] (* G. C. Greubel, May 17 2017 *)
Table[2 Cos[n Pi/2] + RootSum[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, #^n &], {n, 20}] (* Eric W. Weisstein, May 26 2017 *)

PARI

Vec((-7*x^6-6*x^5+5*x^4+4*x^3+15*x^2+2*x+3)/((x^2+1)*(x^5+x^4-2*x^3-2*x^2-3*x+1))+O(x^15)) \\ Andrew Howroyd, May 10 2017

Formula

From Andrew Howroyd, May 10 2017: (Start)

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

G.f.: x*(-7*x^6 - 6*x^5 + 5*x^4 + 4*x^3 + 15*x^2 + 2*x + 3)/((x^2 + 1)*(x^5 + x^4 - 2*x^3 - 2*x^2 - 3*x + 1)). (End)

Extensions

a(1)-a(2) and a(16)-a(25) from Andrew Howroyd, May 10 2017