|
%I #22 Sep 08 2022 08:46:19
%S 2,6,7,18,39,75,155,310,638,1295,2624,5339,10853,22069,44836,91134,
%T 185259,376542,765331,1555567,3161843,6426646,13062506,26550391,
%U 53965428,109688223,222948193,453156469,921069708,1872133138,3805230243,7734373962,15720610559
%N Number of minimal dominating sets in the n-ladder graph.
%H Andrew Howroyd, <a href="/A290379/b290379.txt">Table of n, a(n) for n = 1..200</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LadderGraph.html">Ladder Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MinimalDominatingSet.html">Minimal Dominating Set</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (0, 1, 3, 4, 4, 1, 2, 3, 5, 4, 2).
%F From _Andrew Howroyd_, Aug 01 2017: (Start)
%F 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.
%F 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).
%F (End)
%t 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 *)
%t 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 *)
%t 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 *)
%o (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
%o (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
%Y Row 2 of A286847.
%Y Cf. A290336, A290337.
%K nonn
%O 1,1
%A _Eric W. Weisstein_, Jul 28 2017
%E Terms a(9) and beyond from _Andrew Howroyd_, Aug 01 2017
|
