A303006 Number of minimal total dominating sets in the n-prism graph.

2, 4, 5, 36, 27, 25, 114, 196, 437, 729, 1674, 3249, 6450, 12996, 24870, 49284, 95882, 190969, 369666, 724201, 1425261, 2802276, 5495162, 10764961, 21186827, 41602500, 81686669, 160326244, 314946266, 618516900, 1214288106, 2384368900, 4681737021, 9193357924

Offset

1,1

Comments

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, Apr 17 2018

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Eric Weisstein's World of Mathematics, Prism Graph

Eric Weisstein's World of Mathematics, Total Dominating Set

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).

Formula

G.f.: x*(2 + x^2 + 28*x^3 - 55*x^4 + 26*x^5 + 8*x^6 - 192*x^7 + 359*x^8 - 180*x^9 + 83*x^10 + 552*x^11 - 700*x^12 + 906*x^13 - 583*x^14 - 228*x^15 + 605*x^16 - 1362*x^17 + 596*x^18 - 636*x^19 - 673*x^20 + 684*x^21 - 1045*x^22 + 564*x^23 + 8*x^24 - 154*x^25 + 197*x^26 - 116*x^27 - 107*x^28 + 72*x^29 - 70*x^30 + 36*x^31)/((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 17 2018

Mathematica

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}, {2, 4, 5, 36, 27, 25, 114, 196, 437, 729, 1674, 3249,
  6450, 12996, 24870, 49284, 95882, 190969, 369666, 724201, 1425261,
  2802276, 5495162, 10764961, 21186827, 41602500, 81686669, 160326244, 314946266, 618516900, 1214288106, 2384368900}, 200]
CoefficientList[Series[(2 + x^2 + 28 x^3 - 55 x^4 + 26 x^5 + 8 x^6 - 192 x^7 + 359 x^8 - 180 x^9 + 83 x^10 + 552 x^11 - 700 x^12 + 906 x^13 - 583 x^14 - 228 x^15 + 605 x^16 - 1362 x^17 + 596 x^18 - 636 x^19 - 673 x^20 + 684 x^21 - 1045 x^22 + 564 x^23 + 8 x^24 - 154 x^25 + 197 x^26 - 116 x^27 - 107 x^28 + 72 x^29 - 70 x^30 + 36 x^31)/((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, 199}], x]

Crossrefs

Cf. A290336, A296102, A302405, A303053.

Sequence in context: A126666 A036983 A370438 * A154775 A210418 A167454

Adjacent sequences: A303003 A303004 A303005 * A303007 A303008 A303009

Keyword

nonn,easy

Author

Eric W. Weisstein, Apr 17 2018

Extensions

a(1)-a(2) and terms a(10) and beyond from Andrew Howroyd, Apr 17 2018

Status

approved