A284663 -id:A284663 - 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

A290337 Number of minimal dominating sets in the n-Moebius ladder.

Original entry on oeis.org

2, 4, 11, 28, 37, 67, 149, 284, 596, 1179, 2444, 5023, 10103, 20577, 41746, 84860, 172501, 350392, 712597, 1448463, 2943959, 5983960, 12162310, 24721031, 50246512, 102128407, 207584129, 421927877, 857596064, 1743119352, 3543000201, 7201377180, 14637255611

Offset: 1

Eric W. Weisstein, Jul 27 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Minimal Dominating Set
Eric Weisstein's World of Mathematics, Moebius Ladder

Cf. A284663, A290336, A290379.

Empirical: a(n) = a(n-1)+a(n-2)+2*a(n-3) -a(n-4)+2*a(n-5)-2*a(n-6) +6*a(n-7)+4*a(n-8)+4*a(n-9) -6*a(n-10)-3*a(n-12) +5*a(n-13)-a(n-14)-2*a(n-15) -5*a(n-16)-2*a(n-17)-2*a(n-18) for n > 18. - Andrew Howroyd, Aug 01 2017

Empirical g.f.: x*(2 + 2*x + 5*x^2 + 9*x^3 - 8*x^4 - 20*x^5 - 4*x^6 - 4*x^7 - 40*x^9 - 26*x^10 - 26*x^11 + 14*x^12 - 22*x^13 - 33*x^14 - 45*x^15 - 14*x^16 - 14*x^17) / ((1 - x)*(1 + 2*x^4 + x^6)*(1 - x^2 - 3*x^3 - 4*x^4 - 4*x^5 - x^6 - 2*x^7 - 3*x^8 - 5*x^9 - 4*x^10 - 2*x^11)). - Colin Barker, Aug 02 2017

a(1)-a(2) and terms a(9) and beyond from Andrew Howroyd, Aug 01 2017

A218348 Hilltop maps: number of n X 2 binary arrays indicating the locations of corresponding elements not exceeded by any horizontal or vertical neighbor in a random 0..1 nX2 array.

Original entry on oeis.org

3, 11, 41, 149, 547, 2007, 7361, 27001, 99043, 363299, 1332617, 4888173, 17930307, 65770159, 241251521, 884934705, 3246028995, 11906758971, 43675182633, 160204937605, 587647732323, 2155550649479, 7906775346689, 29002842683433

Offset: 1

R. H. Hardin, Oct 26 2012

Number of dominating sets in the ladder graph P_2 X P_n. - Andrew Howroyd, May 10 2017

			Some solutions for n=3
..1..0....1..1....0..1....1..0....1..1....1..1....0..1....0..0....1..0....0..1
..1..0....0..0....1..1....1..0....0..1....1..1....0..1....1..1....0..0....0..0
..0..1....1..1....0..1....1..0....1..0....1..0....1..0....0..0....0..1....1..0

R. H. Hardin, Table of n, a(n) for n = 1..210
Eric Weisstein's World of Mathematics, Dominating Set
Eric Weisstein's World of Mathematics, Ladder Graph
Index entries for linear recurrences with constant coefficients, signature (3,2,2,-1,-1).

Column 2 of A218354.

Cf. A284663, A284702.

LinearRecurrence[{3, 2, 2, -1, -1}, {3, 11, 41, 149, 547}, 20]  (* Eric W. Weisstein, Jun 14 2017 *)
CoefficientList[Series[(x (3 + 2 x + 2 x^2 - 2 x^3 - x^4))/(1 - 3 x - 2 x^2 - 2 x^3 + x^4 + x^5), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 14 2017 *)
Table[RootSum[1 + # - 2 #^2 - 2 #^3 - 3 #^4 + #^5 &, (-167 + 525 # - 73 #^2 + 819 #^3 - 218 #^4) #^n &]/2102, {n, 20}] (* Eric W. Weisstein, Jul 13 2017 *)

Vec((3+2*x+2*x^2-2*x^3-x^4)/(1-3*x-2*x^2-2*x^3+x^4+x^5)+O(x^50)) \\ Andrew Howroyd, May 10 2017

a(n) = 3*a(n-1) +2*a(n-2) +2*a(n-3) -a(n-4) -a(n-5).

G.f.: x*(3 + 2*x + 2*x^2 - 2*x^3 - x^4)/(1 - 3*x - 2*x^2 - 2*x^3 + x^4 + x^5). - Andrew Howroyd, May 10 2017

A295420 Number of total dominating sets in the n-Moebius ladder.

Original entry on oeis.org

1, 11, 49, 131, 441, 1499, 5041, 17155, 58081, 196331, 664225, 2246915, 7601049, 25714875, 86992929, 294294531, 995591809, 3368061131, 11394068049, 38545859971, 130399709881, 441139059867, 1492362754129, 5048627019523, 17079382863841, 57779138374059

Offset: 1

Eric W. Weisstein, Apr 16 2018

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, Apr 16 2018

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Moebius Ladder
Eric Weisstein's World of Mathematics, Total Dominating Set
Index entries for linear recurrences with constant coefficients, signature (3,0,4,-2,10,4,0,-1,-1).

Cf. A284663, A290337.

Table[RootSum[-1 - # - 3 #^2 + #^3 &, #^n &] - RootSum[1 + # - #^2 + #^3 &, #^n &] + RootSum[-1 + # + #^2 + #^3 &, #^n &], {n, 20}]
LinearRecurrence[{3, 0, 4, -2, 10, 4, 0, -1, -1}, {1, 11, 49, 131, 441, 1499, 5041, 17155, 58081}, 20]
CoefficientList[Series[(1 + 8 x + 16 x^2 - 20 x^3 + 6 x^4 - 8 x^5 + 4 x^6 - 4 x^7 - 3 x^8)/(1 - 3 x - 4 x^3 + 2 x^4 - 10 x^5 - 4 x^6 + x^8 + x^9), {x, 0, 20}], x]

Vec((1 - x)*(1 + 9*x + 25*x^2 + 5*x^3 + 11*x^4 + 3*x^5 + 7*x^6 + 3*x^7)/((1 - x + x^2 + x^3)*(1 + x + x^2 - x^3)*(1 - 3*x - x^2 - x^3)) + O(x^30)) \\ Andrew Howroyd, Apr 16 2018

From Andrew Howroyd, Apr 16 2018: (Start)
G.f.: x*(1 - x)*(1 + 9*x + 25*x^2 + 5*x^3 + 11*x^4 + 3*x^5 + 7*x^6 + 3*x^7)/((1 - x + x^2 + x^3)*(1 + x + x^2 - x^3)*(1 - 3*x - x^2 - x^3)).
a(n) = 3*a(n-1) + 4*a(n-3) - 2*a(n-4) + 10*a(n-5) + 4*a(n-6) - a(n-8) - a(n-9) for n > 9. (End)

a(1)-a(2) and terms a(10) and beyond from Andrew Howroyd, Apr 16 2018

A290509 Number of irredundant sets in the n-Moebius ladder.

Original entry on oeis.org

3, 5, 24, 81, 198, 542, 1550, 4497, 12732, 36210, 103579, 296250, 846303, 2417308, 6907329, 19737889, 56396602, 161138306, 460420182, 1315565326, 3758963099, 10740463167, 30688703123, 87686813826, 250547405698, 715888629071, 2045507376543, 5844625042904

Offset: 1

Eric W. Weisstein, Aug 04 2017

nonn

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

Cf. A284663, A290337.

Empirical: a(n) = 3*a(n-1) - a(n-3) + 2*a(n-4) - 6*a(n-5) - 8*a(n-6) + 13*a(n-7) + 6*a(n-8) - 6*a(n-9) - 2*a(n-10) - 5*a(n-11) - 4*a(n-12) + 2*a(n-13) + 2*a(n-14) - a(n-15) + a(n-16) + 3*a(n-17) + 2*a(n-19) for n > 19. - Andrew Howroyd, Aug 05 2017

Empirical g.f.: x*(3 - 4*x + 9*x^2 + 12*x^3 - 46*x^4 - 20*x^5 + 11*x^6 + 28*x^7 - 18*x^8 - 8*x^9 + 21*x^10 - 4*x^11 + 18*x^12 + 8*x^13 - 15*x^14 + 4*x^15 + 23*x^16 + 14*x^18) / ((1 - x)*(1 - x^2 + x^4 + x^6)*(1 - 2*x - x^2 - 3*x^3 - 5*x^4 + 2*x^5 + 6*x^6 + 5*x^7 + 4*x^8 + 4*x^9 + 3*x^10 + 2*x^11 + 2*x^12)). - Colin Barker, Aug 05 2017

a(1)-a(2) and terms a(10) and beyond from Andrew Howroyd, Aug 05 2017

A303162 Number of minimal total dominating sets in the n-Moebius ladder.

Original entry on oeis.org

0, 6, 9, 14, 25, 57, 196, 222, 441, 851, 1936, 3281, 6084, 12662, 24964, 48830, 93636, 188265, 369664, 725859, 1423249, 2798582, 5503716, 10790049, 21206025, 41601462, 81703521, 160396110, 314991504, 618413702, 1214104336, 2384319102, 4681706929, 9192838950

Offset: 1

Eric W. Weisstein, Apr 19 2018

Sequence extrapolated to n=1 using recurrence.

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Minimal Total Dominating Set.
Eric Weisstein's World of Mathematics, Moebius Ladder.
Index entries for linear recurrences with constant coefficients, signature (2, -2, 3, 4, -7, 5, 0, -21, 39, -24, 21, 33, -36, 63, -33, 0, 33, -63, 36, -33, -21, 24, -39, 21, 0, -5, 7, -4, -3, 2, -2, 1).

Cf. A284663, A290337, A295420, A303006, A303046.

Table[3 - 3 (-1)^n + RootSum[1 - 2 # - #^2 + 3 #^3 - #^4 - 2 #^5 + #^6 &, #^n &] - RootSum[1 - 4 # + 10 #^2 - 19 #^3 + 28 #^4 - 34 #^5 + 37 #^6 - 34 #^7 + 28 #^8 - 19 #^9 + 10 #^10 - 4 #^11 + #^12 &, #^n &] + RootSum[1 + 4 # + 10 #^2 + 19 #^3 + 28 #^4 + 34 #^5 + 37 #^6 + 34 #^7 + 28 #^8 + 19 #^9 + 10 #^10 + 4 #^11 + #^12 &, #^n &], {n, 200}]
LinearRecurrence[{2, -2, 3, 4, -7, 5, 0, -21, 39, -24, 21, 33, -36,
  63, -33, 0, 33, -63, 36, -33, -21, 24, -39, 21, 0, -5, 7, -4, -3,
  2, -2, 1}, {0, 6, 9, 14, 25, 57, 196, 222, 441, 851, 1936, 3281,
  6084, 12662, 24964, 48830, 93636, 188265, 369664, 725859, 1423249,
  2798582, 5503716, 10790049, 21206025, 41601462, 81703521, 160396110,314991504, 618413702, 1214104336, 2384319102}, 200]
Rest @ CoefficientList[Series[x^2 (6 - 3 x + 8 x^2 - 3 x^3 - 16 x^4 + 96 x^5 - 154 x^6 + 171 x^7 - 172 x^8 - 105 x^9 + 74 x^10 - 280 x^11 - 8 x^12 + 91 x^13 - 508 x^14 + 289 x^15 - 386 x^16 - 64 x^17 - 124 x^18 - 231 x^19 - 28 x^20 - 63 x^21 - 28 x^22 + 96 x^23 - 46 x^24 + 39 x^25 - 16 x^26 - 21 x^27 + 18 x^28 - 12 x^29 + 6 x^30)/((1 - x) (1 + x) (1 - 2 x - x^2 + 3 x^3 - x^4 - 2 x^5 + x^6) (1 - 4 x + 10 x^2 - 19 x^3 + 28 x^4 - 34 x^5 + 37 x^6 - 34 x^7 + 28 x^8 - 19 x^9 + 10 x^10 - 4 x^11 + x^12) (1 + 4 x + 10 x^2 + 19 x^3 + 28 x^4 + 34 x^5 + 37 x^6 + 34 x^7 + 28 x^8 + 19 x^9 + 10 x^10 + 4 x^11 + x^12)), {x, 0, 200}], x]

G.f.: x^2*(6 - 3*x + 8*x^2 - 3*x^3 - 16*x^4 + 96*x^5 - 154*x^6 + 171*x^7 - 172*x^8 - 105*x^9 + 74*x^10 - 280*x^11 - 8*x^12 + 91*x^13 - 508*x^14 + 289*x^15 - 386*x^16 - 64*x^17 - 124*x^18 - 231*x^19 - 28*x^20 - 63*x^21 - 28*x^22 + 96*x^23 - 46*x^24 + 39*x^25 - 16*x^26 - 21*x^27 + 18*x^28 - 12*x^29 + 6*x^30)/((1 - x)*(1 + x)*(1 - 2*x - x^2 + 3*x^3 - x^4 - 2*x^5 + x^6)*(1 - 4*x + 10*x^2 - 19*x^3 + 28*x^4 - 34*x^5 + 37*x^6 - 34*x^7 + 28*x^8 - 19*x^9 + 10*x^10 - 4*x^11 + x^12)*(1 + 4*x + 10*x^2 + 19*x^3 + 28*x^4 + 34*x^5 + 37*x^6 + 34*x^7 + 28*x^8 + 19*x^9 + 10*x^10 + 4*x^11 + x^12)). - Andrew Howroyd, Apr 19 2018

a(1)-a(2) and terms a(11) and beyond from Andrew Howroyd, Apr 19 2018

A347559 Number of minimum dominating sets in the n-Moebius ladder.

Original entry on oeis.org

9, 24, 10, 4, 14, 80, 18, 4, 22, 168, 26, 4, 30, 288, 34, 4, 38, 440, 42, 4, 46, 624, 50, 4, 54, 840, 58, 4, 62, 1088, 66, 4, 70, 1368, 74, 4, 78, 1680, 82, 4, 86, 2024, 90, 4, 94, 2400, 98, 4, 102, 2808, 106, 4, 110, 3248, 114, 4, 118, 3720, 122, 4, 126, 4224

Offset: 3

Eric W. Weisstein, Sep 06 2021

Eric Weisstein's World of Mathematics, Minimum Dominating Set
Eric Weisstein's World of Mathematics, Moebius Ladder
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Cf. A284663, A290337.

Table[Piecewise[{{9, n == 3}, {n (n + 2), Mod[n, 4] == 0}, {2 n, Mod[n, 2] == 1}, {4, Mod[n, 4] == 2}}, 0], {n, 3, 20}]
Join[{9}, LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {24, 10, 4, 14, 80, 18, 4, 22, 168, 26, 4, 30}, 20]]
CoefficientList[Series[(-9 - 24 x - 10 x^2 - 4 x^3 + 13 x^4 - 8 x^5 + 12 x^6 + 8 x^7 - 7 x^8 - 2 x^10 - 4 x^11 + 3 x^12)/((-1 + x)^3 (1 + x)^3 (1 + x^2)^3), {x, 0, 20}], x]

a(n) = n*(n+2) for n == 0 (mod 4).
a(n) = 2*n for n == 1 (mod 2) and n > 3.
a(n) = 4 for n == 2 (mod 4).
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12) for n > 3.
G.f.: x^3*(-9 - 24*x - 10*x^2 - 4*x^3 + 13*x^4 - 8*x^5 + 12*x^6 + 8*x^7 - 7*x^8 - 2*x^10 - 4*x^11 + 3*x^12)/((-1 + x)^3*(1 + x)^3*(1 + x^2)^3).

cp's OEIS Frontend

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

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A290337 Number of minimal dominating sets in the n-Moebius ladder.

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A218348 Hilltop maps: number of n X 2 binary arrays indicating the locations of corresponding elements not exceeded by any horizontal or vertical neighbor in a random 0..1 nX2 array.

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A295420 Number of total dominating sets in the n-Moebius ladder.

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A290509 Number of irredundant sets in the n-Moebius ladder.

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A303162 Number of minimal total dominating sets in the n-Moebius ladder.

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A347559 Number of minimum dominating sets in the n-Moebius ladder.

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