A303053 -id:A303053 - cp's OEIS frontend

A302405 Total domination number of the n-prism graph.

0, 1, 2, 2, 4, 4, 4, 5, 6, 6, 8, 8, 8, 9, 10, 10, 12, 12, 12, 13, 14, 14, 16, 16, 16, 17, 18, 18, 20, 20, 20, 21, 22, 22, 24, 24, 24, 25, 26, 26, 28, 28, 28, 29, 30, 30, 32, 32, 32, 33, 34, 34, 36, 36, 36, 37, 38, 38, 40, 40, 40, 41, 42, 42, 44, 44, 44, 45, 46, 46, 48

Offset: 0

Eric W. Weisstein, Apr 07 2018

Extended to a(0)-a(2) using the formula/recurrence.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Prism Graph
Eric Weisstein's World of Mathematics, Total Domination Number
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Cf. A296102, A303006, A303053.

I:=[1,2,2,4,4,4,5,6]; [0] cat [n le 7 select I[n] else Self(n-1) + Self(n-6) - Self(n-7): n in [1..30]]; // G. C. Greubel, Apr 09 2018

Table[(3 + (-1)^n + 4 n - Cos[n Pi/3] - 3 Cos[2 n Pi/3] - Sqrt[3] Sin[n Pi/3] + Sin[2 n Pi/3]/Sqrt[3])/6, {n, 0, 50}]
LinearRecurrence[{1,0,0,0,0,1,-1}, {1,2,2,4,4,4,5,6}, {0, 50}]
CoefficientList[Series[x (1 + x + 2 x^3)/((-1 + x)^2 (1 + x + x^2 + x^3 + x^4 + x^5)), {x, 0, 50}], x]

x='x+O('x^50); concat(0, Vec(x*(1+x+2*x^3)/((1-x)^2*(1+x+ x^2+x^3+ x^4+x^5)))) \\ G. C. Greubel, Apr 09 2018

a(n) = a(n-1) + a(n-6) - a(n-7).
G.f.: x*(1 + x + 2*x^3)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5)).
a(n) = a(n-6) + 4. - Andrew Howroyd, Apr 17 2018

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

Original entry on oeis.org

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

Eric W. Weisstein, Apr 17 2018

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

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, Prism Graph.
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. A290336, A296102, A302405, A303053.

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]

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

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

cp's OEIS Frontend

A302405 Total domination number of the n-prism graph.

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