A284699 -id:A284699 - cp's OEIS frontend

A284702 Number of dominating sets in the n-prism graph.

3, 11, 51, 183, 663, 2435, 8935, 32775, 120219, 440971, 1617531, 5933271, 21763823, 79831875, 292831311, 1074134535, 3940032883, 14452434635, 53012975555, 194456895863, 713287340551, 2616409296963, 9597250953527, 35203676264199, 129130605057163

Offset: 1

Eric W. Weisstein, Apr 01 2017

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, May 10 2017

G. C. Greubel, Table of n, a(n) for n = 1..1000 (terms 1..200 from Andrew Howroyd)
Eric Weisstein's World of Mathematics, Dominating Set
Eric Weisstein's World of Mathematics, Prism Graph
Index entries for linear recurrences with constant coefficients, signature (3,1,5,1,1,-1,-1).

Cf. A051927, A102080, A284699, A218348 (ladder), A284663.

LinearRecurrence[{3, 1, 5, 1, 1, -1, -1}, {3, 11, 51, 183, 663, 2435,
  8935}, 20] (* Eric W. Weisstein, May 17 2017 *)
Rest[CoefficientList[Series[x (-7 x^6 - 6 x^5 + 5 x^4 + 4 x^3 + 15 x^2 + 2 x + 3)/((x^2 + 1) (x^5 + x^4 - 2 x^3 - 2 x^2 - 3 x + 1)), {x, 0, 20}], x]] (* G. C. Greubel, May 17 2017 *)
Table[2 Cos[n Pi/2] + RootSum[1 + #1 - 2 #1^2 - 2 #1^3 - 3 #1^4 + #1^5 &, #^n &], {n, 20}] (* Eric W. Weisstein, May 26 2017 *)

Vec((-7*x^6-6*x^5+5*x^4+4*x^3+15*x^2+2*x+3)/((x^2+1)*(x^5+x^4-2*x^3-2*x^2-3*x+1))+O(x^15)) \\ Andrew Howroyd, May 10 2017

From Andrew Howroyd, May 10 2017: (Start)
a(n) = 3*a(n-1) + a(n-2) + 5*a(n-3) + a(n-4) + a(n-5) - a(n-6) - a(n-7).
G.f.: x*(-7*x^6 - 6*x^5 + 5*x^4 + 4*x^3 + 15*x^2 + 2*x + 3)/((x^2 + 1)*(x^5 + x^4 - 2*x^3 - 2*x^2 - 3*x + 1)). (End)

a(1)-a(2) and a(16)-a(25) from Andrew Howroyd, May 10 2017

A284700 Number of edge covers in the n-antiprism graph.

Original entry on oeis.org

4, 13, 205, 2902, 41413, 590758, 8427370, 120219259, 1714968133, 24464596729, 348995693650, 4978540849669, 71020558255594, 1013132129923498, 14452670295681235, 206172198577335937, 2941115696724530533, 41956003773586931038, 598516493115066264085

Offset: 0

Eric W. Weisstein, Apr 01 2017

nonn

Sequence extrapolated to n=0 using recurrence. - Andrew Howroyd, May 15 2017

G. C. Greubel, Table of n, a(n) for n = 0..860 (terms 0..200 from Andrew Howroyd)
Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Edge Cover
Index entries for linear recurrences with constant coefficients, signature (13,18,1,-4).

Cf. A123304, A020866, A192742, A286910, A284699.

Table[RootSum[4 - # - 18 #^2 - 13 #^3 + #^4 &, #^n &], {n, 0, 20}] (* Eric W. Weisstein, May 17 2017 *)
LinearRecurrence[{13, 18, 1, -4}, {13, 205, 2902, 41413}, {0, 20}] (* Eric W. Weisstein, May 17 2017 *)
CoefficientList[Series[(-x^3-36*x^2-39*x+4)/(4*x^4-x^3-18*x^2-13*x+1), {x, 0, 50}], x]

Vec((-x^3-36*x^2-39*x+4)/(4*x^4-x^3-18*x^2-13*x+1)+O(x^20)) \\ Andrew Howroyd, May 15 2017

From Andrew Howroyd, May 15 2017 (Start)
a(n) = 13*a(n-1)+18*a(n-2)+a(n-3)-4*a(n-4) for n>=4.
G.f.: (-x^3-36*x^2-39*x+4)/(4*x^4-x^3-18*x^2-13*x+1).
(End)

a(0)-a(2) and a(9)-a(18) from Andrew Howroyd, May 15 2017

A290377 Number of minimal dominating sets in the n-antiprism graph.

Original entry on oeis.org

4, 15, 12, 25, 55, 112, 188, 438, 789, 1573, 3135, 5980, 11848, 23035, 45020, 87873, 171910, 335464, 655397, 1281190, 2501173, 4888098, 9548543, 18653025, 36441500, 71190933, 139076320, 271694910, 530784135, 1036914040, 2025703900, 3957367099, 7731003525

Offset: 2

Eric W. Weisstein, Jul 28 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 2..200
Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Minimal Dominating Set

Cf. A284699, A290336.

Empirical: a(n) = 2*a(n-2)+5*a(n-3)+a(n-4) -5*a(n-5)-8*a(n-6)+a(n-7) +6*a(n-8)+10*a(n-9)-2*a(n-10) -2*a(n-11)-5*a(n-12)+a(n-15) for n>16. - Andrew Howroyd, Aug 01 2017

Empirical g.f.: x^2*(4 + 15*x + 4*x^2 - 25*x^3 - 48*x^4 + 7*x^5 + 48*x^6 + 90*x^7 - 20*x^8 - 22*x^9 - 60*x^10 + 15*x^13) / (1 - 2*x^2 - 5*x^3 - x^4 + 5*x^5 + 8*x^6 - x^7 - 6*x^8 - 10*x^9 + 2*x^10 + 2*x^11 + 5*x^12 - x^15). - Colin Barker, Aug 01 2017

a(2) and terms a(8) and beyond from Andrew Howroyd, Aug 01 2017

A338154 a(n) is the number of acyclic orientations of the edges of the n-antiprism.

Original entry on oeis.org

426, 4968, 50640, 486930, 4547088, 41796168, 380789562, 3451622904, 31194607488, 281440825122, 2536622917920, 22848990484344, 205743704494026, 1852238413383048, 16673036119790640, 150072652217086770, 1350735146332489008, 12157047307392618408

Offset: 3

Peter Kagey, Oct 13 2020

nonn

Conjectured linear recurrence and g.f. confirmed by Kagey's formula. - Ray Chandler, Mar 10 2024

			For n = 3, the 3-antiprism is the octahedron (3-dimensional cross-polytope), so a(3) = A033815(3) = 426.

Peter Kagey, Table of n, a(n) for n = 3..1000
Eric Weisstein's World of Mathematics, Antiprism Graph
Wikipedia, Acyclic orientation
Index entries for linear recurrences with constant coefficients, signature (17, -88, 153, -81).

Cf. A077263, A124352, A124353, A192742, A284699, A284700, A284701, A287988, A294152, A297384.

Cf. A033815 (cross-polytope), A058809 (wheel), A334247 (hypercube), A338152 (demihypercube), A338153 (prism).

A338154[n_] := Round[-2^(1-n)*((7 - Sqrt[13])^n + (7 + Sqrt[13])^n) + 9^n + 5] (* Peter Kagey, Nov 15 2020 *)

Conjectures from Colin Barker, Oct 13 2020: (Start)
G.f.: 6*x^3*(71 - 379*x + 612*x^2 - 324*x^3) / ((1 - x)*(1 - 9*x)*(1 - 7*x + 9*x^2)).
a(n) = 17*a(n-1) - 88*a(n-2) + 153*a(n-3) - 81*a(n-4) for n>6.
(End)
a(n) = -2^(1-n)*((7-sqrt(13))^n + (7+sqrt(13))^n) + 9^n + 5. - Peter Kagey, Nov 15 2020

A290510 Number of irredundant sets in the n-antiprism graph.

Original entry on oeis.org

1, 5, 22, 37, 76, 194, 491, 1125, 2731, 6640, 15962, 38386, 92639, 223403, 538102, 1297061, 3126726, 7535759, 18162481, 43777272, 105514634, 254314406, 612962766, 1477397778, 3560896401, 8582652759, 20686361179, 49859334611, 120173522734, 289648431514

Offset: 1

Eric W. Weisstein, Aug 04 2017

nonn

The n-antiprism graphs are well defined for n>=3. Sequence extendend to n=1 using recurrence. - Andrew Howroyd, Aug 05 2017

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Irredundant Set

Cf. A284699, A290377.

Empirical: a(n) = a(n-1)+2*a(n-2)+5*a(n-3) -6*a(n-5)-10*a(n-6) +6*a(n-8)+10*a(n-9)-a(n-10) -2*a(n-11)-5*a(n-12)+a(n-15) for n>15. - Andrew Howroyd, Aug 05 2017

Empirical g.f.: x*(1 + 4*x + 15*x^2 - 30*x^4 - 60*x^5 + 48*x^7 + 90*x^8 - 10*x^9 - 22*x^10 - 60*x^11 + 15*x^14) / ((1 - x)*(1 + x + x^2)*(1 - x - 2*x^2 - 4*x^3 - x^4 + 4*x^5 + 6*x^6 - x^7 - 2*x^8 - 4*x^9 + x^12)). - Colin Barker, Aug 05 2017

a(1)-a(2) and terms a(11) and beyond from Andrew Howroyd, Aug 05 2017

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

Original entry on oeis.org

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

Eric W. Weisstein, May 14 2018

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

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).

Cf. A284699, A290377.

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]

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

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

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

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

cp's OEIS Frontend

A284702 Number of dominating sets in the n-prism graph.

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A284700 Number of edge covers in the n-antiprism graph.

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A290377 Number of minimal dominating sets in the n-antiprism graph.

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A338154 a(n) is the number of acyclic orientations of the edges of the n-antiprism.

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A290510 Number of irredundant sets in the n-antiprism graph.

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

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