A284663 - OEIS

%I #27 Feb 16 2025 08:33:43

%S 3,15,51,179,663,2439,8935,32771,120219,440975,1617531,5933267,

%T 21763823,79831879,292831311,1074134531,3940032883,14452434639,

%U 53012975555,194456895859,713287340551,2616409296967,9597250953527,35203676264195,129130605057163

%N Number of dominating sets in the Moebius ladder M_n.

%C Sequence extrapolated to n=1 using recurrence. - _Andrew Howroyd_, May 10 2017

%H Andrew Howroyd, <a href="/A284663/b284663.txt">Table of n, a(n) for n = 1..200</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DominatingSet.html">Dominating Set</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MoebiusLadder.html">Moebius Ladder</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,5,1,1,-1,-1).

%F From _Andrew Howroyd_, May 10 2017 (Start)

%F 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) for n>7.

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

%t LinearRecurrence[{3, 1, 5, 1, 1, -1, -1}, {3, 15, 51, 179, 663, 2439,

%t   8935}, 20] (* _Eric W. Weisstein_, May 17 2017 *)

%t Rest[CoefficientList[Series[x*(1 - x)*(1 + x)*(3*x^4 + 2*x^3 + 6*x^2 + 6*x + 3)/((x^2 + 1)*(x^5 + x^4 - 2*x^3 - 2*x^2 - 3*x + 1)), {x,0,50}], x]] (* _G. C. Greubel_, May 17 2017 *)

%t Table[RootSum[1 + # - 2 #^2 - 2 #^3 - 3 #^4 + #^5 &, #^n &] - 2 Cos[n Pi/2], {n, 20}] (* _Eric W. Weisstein_, Jun 14 2017 *)

%o (PARI)

%o Vec((1-x)*(1+x)*(3*x^4+2*x^3+6*x^2+6*x+3)/((x^2+1)*(x^5+x^4-2*x^3-2*x^2-3*x+1))+O(x^50)) \\ _Andrew Howroyd_, May 10 2017

%Y Cf. A182143, A284702, A218348 (ladder).

%K nonn

%O 1,1

%A _Eric W. Weisstein_, Mar 31 2017

%E a(1)-(2) and a(16)-a(25) from _Andrew Howroyd_, May 10 2017