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A302255 Total domination number of the n-antiprism graph. 3

%I

%S 0,1,2,2,3,4,4,4,5,6,6,7,8,8,8,9,10,10,11,12,12,12,13,14,14,15,16,16,

%T 16,17,18,18,19,20,20,20,21,22,22,23,24,24,24,25,26,26,27,28,28,28,29,

%U 30,30,31,32,32,32,33,34,34,35,36,36,36,37,38,38,39,40,40,40,41

%N Total domination number of the n-antiprism graph.

%C Sequence extended to a(0)-a(2) using the recurrence/formula.

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotalDominationNumber.html">Total Domination Number</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,1,-1).

%F a(n) = a(n-1) + a(n-7) - a(n-8).

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

%F a(n) = a(n-7) + 4. - _Andrew Howroyd_, Apr 18 2018

%F a(n) = a(n-7*k) + 4*k. - _Eric W. Weisstein_, Apr 19 2018

%t Table[(4 + 4 n + E^(4 I n Pi/7) Root[1 + 7 #^2 + 28 #^3 + 42 #^4 + 28 #^5 + 7 #^6 &, 1] + E^(-4 I n Pi/7) Root[1 + 7 #^2 + 28 #^3 + 42 #^4 + 28 #^5 + 7 #^6 &, 2] + E^(-2 I n Pi/7) Root[1 + 7 #^2 + 28 #^3 + 42 #^4 + 28 #^5 + 7 #^6 &, 3] + E^(2 I n Pi/7) Root[1 + 7 #^2 + 28 #^3 + 42 #^4 + 28 #^5 + 7 #^6 &, 4] + E^(-6 I n Pi/7) Root[1 + 7 #^2 + 28 #^3 + 42 #^4 + 28 #^5 + 7 #^6 &, 5] + E^(6 I n Pi/7) Root[1 + 7 #^2 + 28 #^3 + 42 #^4 + 28 #^5 + 7 #^6 &, 6])/ 7, {n, 20}] // RootReduce

%t LinearRecurrence[{1,0,0,0,0,0,1,-1}, {1,2,2,3,4,4,4,5,6,6}, {0, 20}]

%t CoefficientList[Series[x (1 + x + x^3 + x^4)/((1 - x)^2 (1 + x + x^2 + x^3 + x^4 + x^5 + x^6)), {x, 0, 20}], x]

%o (PARI) x='x+O('x^50); concat(0, Vec(x*(1+x+x^3+x^4)/((1-x)^2*(1+x+x^2+ x^3+x^4+x^5+x^6)))) \\ _G. C. Greubel_, Apr 09 2018

%o (MAGMA) I:=[2,2,3,4,4,4,5,6]; [0,1] cat [n le 8 select I[n] else Self(n-1) + Self(n-7) - Self(n-8): n in [1..30]]; // _G. C. Greubel_, Apr 09 2018

%Y Cf. A302652, A302760, A302763.

%K nonn,easy

%O 0,3

%A _Eric W. Weisstein_, Apr 07 2018

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Last modified July 30 05:57 EDT 2021. Contains 346349 sequences. (Running on oeis4.)