A290513 Number of irredundant sets in the n-ladder graph.

3, 11, 26, 79, 224, 640, 1828, 5225, 14928, 42654, 121873, 348232, 995003, 2843014, 8123337, 23210809, 66320216, 189496620, 541448364, 1547079580, 4420468031, 12630596045, 36089381477, 103118131368, 294639269914, 841872308017, 2405480380385, 6873175192304

Offset: 1

Author

Eric W. Weisstein, Aug 04 2017

Keywords

• nonn

Comments

Row 2 of A286868.

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Irredundant Set
• Eric Weisstein's World of Mathematics, Ladder Graph
• Index entries for linear recurrences with constant coefficients, signature (2, 1, 3, 5, -2, -6, -5, -4, -4, -3, -2, -2).

Crossrefs

Cf. A286868 (irredundant sets in m X n grid graph).

Programs

Mathematica

Table[RootSum[2 + 2 # + 3 #^2 + 4 #^3 + 4 #^4 + 5 #^5 + 6 #^6 + 2 #^7 - 5 #^8 - 3 #^9 - #^10 - 2 #^11 + #^12 &, -105577159431355949 #^n - 389671420034091247 #^(n + 1) + 29241021604101932 #^(n + 2) + 551171239538727862 #^(n + 3) - 835076245333578054 #^(n + 4) - 823168743731791895 #^(n + 5) + 1288188291747539683 #^(n + 6) + 114497498217658607 #^(n + 7) - 529545500369064866 #^(n + 8) + 763912087212707104 #^(n + 9) - 593709228550713556 #^(n + 10) + 130400048784652699 #^(n + 11) &]/2485579507903393779, {n, 20}] (* Eric W. Weisstein, Aug 05 2017 *)
LinearRecurrence[{2, 1, 3, 5, -2, -6, -5, -4, -4, -3, -2, -2}, {3, 11, 26, 79, 224, 640, 1828, 5225, 14928, 42654, 121873, 348232}, 20] (* Eric W. Weisstein, Aug 05 2017 *)
CoefficientList[Series[(3 + 5 x + x^2 + 7 x^3 - 8 x^4 - 14 x^5 - 3 x^6 - 5 x^7 - 9 x^8 - 3 x^9 - 2 x^10 - 2 x^11)/(1 - 2 x - x^2 - 3 x^3 - 5 x^4 + 2 x^5 + 6 x^6 + 5 x^7 + 4 x^8 + 4 x^9 + 3 x^10 + 2 x^11 + 2 x^12), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 05 2017 *)

PARI

Vec((3+5*x+x^2+7*x^3-8*x^4-14*x^5-3*x^6-5*x^7-9*x^8-3*x^9-2*x^10-2*x^11)/(1-2*x-x^2-3*x^3-5*x^4+2*x^5+6*x^6+5*x^7+4*x^8+4*x^9+3*x^10+2*x^11+2*x^12)+O(x^30)) \\ Andrew Howroyd, Aug 04 2017

Formula

From Andrew Howroyd, Aug 04 2017: (Start)

a(n) = 2*a(n-1) + a(n-2) + 3*a(n-3) + 5*a(n-4) - 2*a(n-5) - 6*a(n-6) - 5*a(n-7) - 4*a(n-8) - 4*a(n-9) - 3*a(n-10) - 2*a(n-11) - 2*a(n-12).

G.f.: x*(3 + 5*x + x^2 + 7*x^3 - 8*x^4 - 14*x^5 - 3*x^6 - 5*x^7 - 9*x^8 - 3*x^9 - 2*x^10 - 2*x^11) / (1 - 2*x - x^2 - 3*x^3 - 5*x^4 + 2*x^5 + 6*x^6 + 5*x^7 + 4*x^8 + 4*x^9 + 3*x^10 + 2*x^11 + 2*x^12).

(End)

Extensions

Terms a(11) and beyond from Andrew Howroyd, Aug 04 2017