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A303053
Number of minimum total dominating sets in the n-prism graph.
3
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
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
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