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A304567
Number of maximum irredundant sets in the n-antiprism graph.
1
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
1,2
COMMENTS
Sequence extrapolated to n=1 using formula. - Andrew Howroyd, May 20 2018
LINKS
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).
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
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]
PROG
(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
CROSSREFS
Sequence in context: A366869 A267103 A030553 * A154493 A337519 A031012
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, May 14 2018
EXTENSIONS
a(1)-a(2) and terms a(14) and beyond from Andrew Howroyd, May 20 2018
STATUS
approved