A303053 Number of minimum total dominating sets in the n-prism graph.

2, 4, 3, 36, 25, 9, 14, 64, 3, 625, 99, 9, 26, 196, 3, 3136, 221, 9, 38, 400, 3, 9801, 391, 9, 50, 676, 3, 23716, 609, 9, 62, 1024, 3, 48841, 875, 9, 74, 1444, 3, 90000, 1189, 9, 86, 1936, 3, 152881, 1551, 9, 98, 2500, 3, 244036, 1961, 9, 110, 3136, 3, 370881

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 (0,0,0,0,0,5,0,0,0,0,0,-10,0,0,0,0,0,10,0,0,0,0,0,-5,0,0,0,0,0,1).

Formula

From Andrew Howroyd, Apr 17 2018: (Start)

a(n) = 5*a(n-6) - 10*a(n-12) + 10*a(n-18) - 5*a(n-24) + a(n-30) for n > 30.

a(6*k) = 9, a(6*k+1) = 2*(6*k+1), a(6*k+2) = (6*k+2)^2, a(6*k+3) = 3, a(6*k+4) = ((2*k + 3)*(3*k + 2))^2, a(6*k+5) = (4*k + 5)*(6*k + 5). (End)

Mathematica

Table[(432 + 132 n + 85 n^2 + 10 n^3 + n^4 + (216 - 132 n + 37 n^2 + 10 n^3 + n^4) (-1)^n +(432 + 132 n - 37 n^2 - 10 n^3 - n^4) Cos[n Pi/3] + (864 - 132 n - 85 n^2 - 10 n^3 - n^4) Cos[2 n Pi/3] +Sqrt[3] (12 n - 13 n^2 - 10 n^3 - n^4) Sin[n Pi/3] + Sqrt[3] (12 n - 35 n^2 + 10 n^3 + n^4) Sin[2 n Pi/3])/216, {n, 200}]
Table[Piecewise[{{9, Mod[n, 6] == 0}, {2 n, Mod[n, 6] == 1}, {n^2, Mod[n, 6] == 2}, {3, Mod[n, 6] == 3}, {n^2 (n + 5)^2/36, Mod[n, 6] == 4}, {n (2 n + 5)/3, Mod[n, 6] == 5}}], {n, 200}]
LinearRecurrence[{0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 1}, {2, 4, 3, 36, 25, 9, 14, 64, 3, 625, 99, 9, 26, 196, 3, 3136, 221, 9, 38, 400, 3, 9801, 391, 9, 50, 676, 3, 23716, 609, 9}, 200]
Rest @ CoefficientList[Series[(9 x^6)/(1 - x^6) - (3 x^3)/(-1 + x^6) + (2 x (1 + 5 x^6))/(-1 + x^6)^2 + (x^5 (-25 - 24 x^6 + x^12))/(-1 + x^6)^3 - (4 x^2 (1 + 13 x^6 + 4 x^12))/(-1 + x^6)^3 - (x^4 (36 + 445 x^6 + 371 x^12 + 11 x^18 + x^24))/(-1 + x^6)^5, {x, 0, 200}], x]

Crossrefs

Cf. A296102, A302405, A303006.

Sequence in context: A356063 A071970 A182103 * A163089 A258371 A111172

Adjacent sequences: A303050 A303051 A303052 * A303054 A303055 A303056

Keyword

nonn,easy

Author

Eric W. Weisstein, Apr 17 2018

Extensions

a(1)-a(2) and terms a(15) and beyond from Andrew Howroyd, Apr 17 2018

Status

approved