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

Author

Eric W. Weisstein, Apr 17 2018

Keywords

• nonn
• easy

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, Minimum Total Dominating Set.
• Eric Weisstein's World of Mathematics, Prism Graph.
• 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).

Crossrefs

Cf. A296102, A302405, A303006.

Programs

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]

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)

Extensions

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