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

Author

Eric W. Weisstein, May 14 2018

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A284699, A290377.

Programs

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]

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

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

Extensions

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