A304567 Number of maximum irredundant sets in the n-antiprism graph.
1, 4, 15, 28, 20, 15, 161, 48, 15, 540, 88, 15, 1365, 140, 15, 2896, 204, 15, 5453, 280, 15, 9416, 368, 15, 15225, 468, 15, 23380, 580, 15, 34441, 704, 15, 49028, 840, 15, 67821, 988, 15, 91560, 1148, 15, 121045, 1320, 15, 157136, 1504, 15, 200753, 1700
Offset: 1
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1000
- Eric Weisstein's World of Mathematics, Antiprism Graph
- Eric Weisstein's World of Mathematics, Maximum Irredundant Vertex Set
- Index entries for linear recurrences with constant coefficients, signature (0,0,5,0,0,-10,0,0,10,0,0,-5,0,0,1).
Programs
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Mathematica
Table[Piecewise[{{15, Mod[n, 3] == 0}, {n (n + 2) (29 + 20 n + 5 n^2)/162, Mod[n, 3] == 1}, {2 n (n + 1)/3, Mod[n, 3] == 2}}], {n, 20}] LinearRecurrence[{0, 0, 5, 0, 0, -10, 0, 0, 10, 0, 0, -5, 0, 0, 1}, {1, 4, 15, 28, 20, 15, 161, 48, 15, 540, 88, 15, 1365, 140, 15}, 20] Table[(2430 + 166 n + 177 n^2 + 30 n^3 + 5 n^4 - (-4860 + 166 n + 177 n^2 + 30 n^3 + 5 n^4) Cos[2 n Pi/3] + Sqrt[3] n (-50 - 39 n + 30 n^2 + 5 n^3) Sin[2 n Pi/3])/486, {n, 20}] CoefficientList[Series[15/(1 - x^3) x^2 - (1 + 23 x^3 + 31 x^6 + 5 x^9)/(-1 + x^3)^5 - (4 x (1 + 2 x^3))/(-1 + x^3)^3, {x, 0, 20}], x] -
PARI
a(n)={if(n%3==0, 15, my(k=n\3); n*(k+1)*if(n%3==1, (5*k^2+10*k+6)/6, 2))} \\ Andrew Howroyd, May 20 2018 -
PARI
Vec(x*(1 + 4*x + 15*x^2 + 23*x^3 - 60*x^5 + 31*x^6 - 12*x^7 + 90*x^8 + 5*x^9 + 8*x^10 - 60*x^11 + 15*x^14) / ((1 - x)^5*(1 + x + x^2)^5) + O(x^50)) \\ Colin Barker, May 22 2018
Formula
From Andrew Howroyd, May 21 2018: (Start)
a(n) = 5*a(n-3) - 10*a(n-6) + 10*a(n-9) - 5*a(n-12) + a(n-15) for n > 15.
a(3*k) = 15, a(3*k+1) = (3*k+1)*(k+1)*(5*k^2+10*k+6)/6, a(3*k+2) = 2*(k+1)*(3*k+2). (End)
G.f.: x*(1 + 4*x + 15*x^2 + 23*x^3 - 60*x^5 + 31*x^6 - 12*x^7 + 90*x^8 + 5*x^9 + 8*x^10 - 60*x^11 + 15*x^14) / ((1 - x)^5*(1 + x + x^2)^5). - Colin Barker, May 22 2018
Extensions
a(1)-a(2) and terms a(14) and beyond from Andrew Howroyd, May 20 2018
Comments