A372712 - OEIS

%I #22 Dec 19 2024 21:15:24

%S 1,16,36,36,100,121,196,324,441,729,1089,1681,2704,4225,6889,11236,

%T 18496,30976,51984,88209,150544,258064,444889,769129,1334025,2319529,

%U 4040100,7049025,12313081,21529600,37675044,65966884,115562500,202521361,355020964,622502500

%N Numbers of minimal total dominating sets in the n-trapezohedral graph.

%H Christian Sievers, <a href="/A372712/b372712.txt">Table of n, a(n) for n = 3..1000</a>

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

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

%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,-4,-1,14,-6,-5,-3,7,-4,3,0,0,-2,1).

%F a(n) = (A001608(n) + n)^2. - _Christian Sievers_, Jun 14 2024

%F From _Eric W. Weisstein_, Dec 12 2024: (Start)

%F a(n) = 4*a(n-1)-3*a(n-2)-4*a(n-3)-a(n-4)+14*a(n-5)-6*a(n-6)-5*a(n-7)-3*a(n-8)+7*a(n-9)-4*a(n-10)+3*a(n-11)-2*a(n-14)+a(n-15).

%F G.f.: -(x*(1+12*x-25*x^2-56*x^3+129*x^4-25*x^5-26*x^6-64*x^7+112*x^8-121*x^9+74*x^10-x^11-7*x^12-14*x^13+9*x^14)/((-1+x)^3*(-1-x+x^3)*(-1+2*x-x^2+x^3)*(-1+x^2+x^3)^2)). (End)

%t Table[(RootSum[-1 - # + #^3 &, #^n &] + n)^2, {n, 20}] (* _Eric W. Weisstein_, Dec 12 2024 *)

%t LinearRecurrence[{4, -3, -4, -1, 14, -6, -5, -3, 7, -4, 3, 0, 0, -2, 1}, {1, 16, 36, 36, 100, 121, 196, 324, 441, 729, 1089, 1681, 2704, 4225, 6889}, 20] (* _Eric W. Weisstein_, Dec 12 2024 *)

%t CoefficientList[Series[-(1 + 12 x - 25 x^2 - 56 x^3 + 129 x^4 - 25 x^5 - 26 x^6 - 64 x^7 + 112 x^8 - 121 x^9 + 74 x^10 - x^11 - 7 x^12 - 14 x^13 + 9 x^14)/((-1 + x)^3 (-1 - x + x^3) (-1 + 2 x - x^2 + x^3) (-1 + x^2 + x^3)^2), {x, 0, 20}], x] (* _Eric W. Weisstein_, Dec 12 2024 *)

%o (PARI) a(n)=(polsym(x^3-x-1,n)[n+1]+n)^2 \\ _Christian Sievers_, Jun 14 2024

%Y Cf. A001608.

%Y Extended to a(1)-a(2) using the formula/recurrence.

%K nonn

%O 1,2

%A _Eric W. Weisstein_, May 11 2024

%E a(13) and beyond from _Christian Sievers_, Jun 14 2024

%E a(1)-a(2) prepended by _Eric W. Weisstein_, Dec 12 2024

%E Offset corrected for above change