A290379 Number of minimal dominating sets in the n-ladder graph.

2, 6, 7, 18, 39, 75, 155, 310, 638, 1295, 2624, 5339, 10853, 22069, 44836, 91134, 185259, 376542, 765331, 1555567, 3161843, 6426646, 13062506, 26550391, 53965428, 109688223, 222948193, 453156469, 921069708, 1872133138, 3805230243, 7734373962, 15720610559

Offset: 1

Author

Eric W. Weisstein, Jul 28 2017

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Ladder Graph
• Eric Weisstein's World of Mathematics, Minimal Dominating Set
• Index entries for linear recurrences with constant coefficients, signature (0, 1, 3, 4, 4, 1, 2, 3, 5, 4, 2).

Crossrefs

Row 2 of A286847.

Cf. A290336, A290337.

Programs

Magma

I:=[2,6,7,18,39,75,155,310,638,1295,2624]; [n le 11 select I[n] else Self(n-2)+3*Self(n-3)+4*Self(n-4)+4*Self(n-5)+Self(n-6)+2*Self(n-7)+3*Self(n-8)+5*Self(n-9)+4*Self(n-10)+2*Self(n-11): n in [1..40]]; // Vincenzo Librandi, Aug 04 2017

Mathematica

Table[-RootSum[-2 - 4 # - 5 #^2 - 3 #^3 - 2 #^4 - #^5 - 4 #^6 - 4 #^7 - 3 #^8 - #^9 + #^11 &, 621827501801 #^n - 301456826961 #^(n + 1) + 280366986955 #^(n + 2) - 1253389979482 #^(n + 3) + 843186094854 #^(n + 4) - 87555893434 #^(n + 5) + 236346312907 #^(n + 6) - 504072574383 #^(n + 7) + 231943645265 #^(n + 8) - 618185916584 #^(n + 9) + 290649224768 #^(n + 10) &]/2097121971853, {n, 20}] (* Eric W. Weisstein, Aug 04 2017 *)
LinearRecurrence[{0, 1, 3, 4, 4, 1, 2, 3, 5, 4, 2}, {2, 6, 7, 18, 39, 75, 155, 310, 638, 1295, 2624}, 20] (* Eric W. Weisstein, Aug 04 2017 *)
CoefficientList[Series[((1 + x) (2 + 4 x + x^2 + 5 x^3 + x^4 + 3 x^5 + 5 x^6 + 3 x^7 + 2 x^8 + 2 x^9))/(1 - x^2 - 3 x^3 - 4 x^4 - 4 x^5 - x^6 - 2 x^7 - 3 x^8 - 5 x^9 - 4 x^10 - 2 x^11), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 04 2017 *)

PARI

Vec((1+x)*(2+4*x+x^2+5*x^3+x^4+3*x^5+5*x^6+3*x^7+2*x^8+2*x^9)/(1-x^2-3*x^3-4*x^4-4*x^5-x^6-2*x^7-3*x^8-5*x^9-4*x^10-2*x^11)+O(x^40)) \\ Andrew Howroyd, Aug 01 2017

Formula

From Andrew Howroyd, Aug 01 2017: (Start)

a(n) = a(n-2) + 3*a(n-3) + 4*a(n-4) + 4*a(n-5) + a(n-6) + 2*a(n-7) + 3*a(n-8) + 5*a(n-9) + 4*a(n-10) + 2*a(n-11) for n > 11.

G.f.: x*(1+x)*(2 + 4*x + x^2 + 5*x^3 + x^4 + 3*x^5 + 5*x^6 + 3*x^7 + 2*x^8 + 2*x^9)/(1 - x^2 - 3*x^3 - 4*x^4 - 4*x^5 - x^6 - 2*x^7- 3*x^8 - 5*x^9 - 4*x^10 - 2*x^11).

(End)

Extensions

Terms a(9) and beyond from Andrew Howroyd, Aug 01 2017