A304568 Number of minimum dominating sets in the n-antiprism graph.

2, 4, 15, 12, 5, 40, 14, 140, 45, 5, 154, 24, 546, 98, 5, 384, 34, 1485, 171, 5, 770, 44, 3289, 264, 5, 1352, 54, 6370, 377, 5, 2170, 64, 11220, 510, 5, 3264, 74, 18411, 663, 5, 4674, 84, 28595, 836, 5, 6440, 94, 42504, 1029, 5, 8602, 104, 60950, 1242, 5

Offset

1,1

Comments

Sequence extrapolated to n=1 using formula. - Andrew Howroyd, May 20 2018

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, Minimum Dominating Set

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,5,0,0,0,0,-10,0,0,0,0,10,0,0,0,0,-5,0,0,0,0,1).

Formula

From Andrew Howroyd, May 20 2018: (Start)

a(n) = 5*a(n-5) - 10*a(n-10) + 10*a(n-15) - 5*a(n-20) + a(n-25) for n > 25.

a(5*k) = 5, a(5*k+1) = 2*(5*k+1)*(k+1)*(2*k+3)/3, a(5*k+2) = 2*(5*k+2), a(5*k+3) = (5*k+3)*(2*k+5)*(2*k+4)*(2*k+3)/12, a(5*k+4)=(5*k+4)*(2*k+3). (End)

G.f.: x*(2 + 4*x + 15*x^2 + 12*x^3 + 5*x^4 + 30*x^5 - 6*x^6 + 65*x^7 - 15*x^8 - 20*x^9 - 26*x^10 - 6*x^11 - 4*x^12 - 7*x^13 + 30*x^14 - 6*x^15 + 14*x^16 + 5*x^17 + 11*x^18 - 20*x^19 - 6*x^21 - x^22 - x^23 + 5*x^24) / ((1 - x)^5*(1 + x + x^2 + x^3 + x^4)^5). - Colin Barker, May 22 2018

Mathematica

Table[Piecewise[{{5, Mod[n, 5] == 0}, {2 n (4 + n) (13 + 2 n)/75, Mod[n, 5] == 1}, {2 n, Mod[n, 5] == 2}, {n (7 + n) (9 + 2 n) (19 + 2 n)/750, Mod[n, 5] == 3}, {n (7 + 2 n)/5, Mod[n, 5] == 4}}], {n, 30}]
LinearRecurrence[{0, 0, 0, 0, 5, 0, 0, 0, 0, -10, 0, 0, 0, 0, 10, 0, 0, 0, 0, -5, 0, 0, 0, 0, 1}, {2, 4, 15, 12, 5, 40, 14, 140, 45, 5, 154, 24, 546, 98, 5, 384, 34, 1485, 171, 5, 770, 44, 3289, 264, 5}, 30]
CoefficientList[Series[(5 x^4)/(1 - x^5) + (2 x (2 + 3 x^5))/(-1 + x^5)^2 + (x^3 (-12 - 9 x^5 + x^10))/(-1 + x^5)^3 + (2 (1 + 16 x^5 + 3 x^10))/(-1 + x^5)^4 + (x^2 (-15 - 65 x^5 + 4 x^10 - 5 x^15 + x^20))/(-1 + x^5)^5, {x, 0, 30}], x]

Prog

(PARI) a(n)={[k->5, k->2*(5*k+1)*(k+1)*(2*k+3)/3, k->2*(5*k+2), k->(5*k+3)*(2*k+5)*(2*k+4)*(2*k+3)/12, k->(5*k+4)*(2*k+3)][n%5+1](n\5)} \\ Andrew Howroyd, May 20 2018
(PARI) Vec(x*(2 + 4*x + 15*x^2 + 12*x^3 + 5*x^4 + 30*x^5 - 6*x^6 + 65*x^7 - 15*x^8 - 20*x^9 - 26*x^10 - 6*x^11 - 4*x^12 - 7*x^13 + 30*x^14 - 6*x^15 + 14*x^16 + 5*x^17 + 11*x^18 - 20*x^19 - 6*x^21 - x^22 - x^23 + 5*x^24) / ((1 - x)^5*(1 + x + x^2 + x^3 + x^4)^5) + O(x^40)) \\ Colin Barker, May 22 2018

Crossrefs

Cf. A284699, A290377.

Sequence in context: A279025 A196239 A019543 * A271645 A265483 A357692

Adjacent sequences: A304565 A304566 A304567 * A304569 A304570 A304571

Keyword

nonn,easy

Author

Eric W. Weisstein, May 14 2018

Extensions

a(1)-a(2) and terms a(21) and beyond from Andrew Howroyd, May 20 2018

Status

approved