A302652 Number of minimum total dominating sets in the n-antiprism graph.

2, 6, 12, 24, 80, 48, 7, 16, 237, 40, 154, 1344, 208, 7, 30, 1136, 68, 396, 6688, 480, 7, 44, 3151, 96, 750, 20800, 864, 7, 58, 6730, 124, 1216, 50160, 1360, 7, 72, 12321, 152, 1794, 103040, 1968, 7, 86, 20372, 180, 2484, 189504, 2688, 7, 100, 31331, 208, 3286

Offset: 1

Author

Eric W. Weisstein, Apr 11 2018

Keywords

• nonn
• easy

Comments

Sequence extrapolated to n=1 using recurrence.

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Antiprism Graph.
• Eric Weisstein's World of Mathematics, Minimum Total Dominating Set.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,5,0,0,0,0,0,0,-10,0,0,0,0,0,0,10,0,0,0,0,0,0,-5,0,0,0,0,0,0,1).

Crossrefs

Cf. A302255, A302760, A302763.

Programs

Mathematica

Table[Piecewise[{{7, Mod[n, 7] == 0}, {2 n, Mod[n, 7] == 1}, {n (37 + 138 n + 32 n^2)/147, Mod[n, 7] == 2}, {4 n, Mod[n, 7] == 3}, {2 n (5 + 4 n)/7, Mod[n, 7] == 4}, {(8 n (2 + n) (9 + n) (1 + 4 n))/1029, Mod[n, 7] == 5}, {8 n (1 + n)/7, Mod[n, 7] == 6}}], {n, 200}]
LinearRecurrence[{0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 1}, {2, 6, 12, 24, 80, 48, 7, 16, 237, 40, 154, 1344, 208, 7, 30, 1136, 68, 396, 6688, 480, 7, 44, 3151, 96, 750, 20800, 864, 7, 58, 6730, 124, 1216, 50160, 1360, 7}, 200]
Rest @ CoefficientList[Series[7 x^7/(1 - x^7) - 16 x^6 (3 + 4 x^7)/(-1 + x^7)^3 + 4 x^3 (3 + 4 x^7)/(-1 + x^7)^2 + 2 x (1 + 6 x^7)/(-1 + x^7)^2 - 2 x^4 (12 + 41 x^7 + 3 x^14)/(-1 + x^7)^3 - 16 x^5 (5 + 59 x^7 + 48 x^14)/(-1 + x^7)^5 + x^2 (6 + 213 x^7 + 224 x^14 + 5 x^21)/(-1 + x^7)^4, {x, 0, 200}], x]

PARI

a(n)={[k->7, k->2*(7*k+1), k->(7*k+2)*(32*k^2+38*k+9)/3, k->4*(7*k+3), k->(7*k+4)*(8*k+6), k->(7*k+5)*(8*k+8)*(k+2)*(4*k+3)/3, k->8*(7*k+6)*(k+1)][1+n%7](n\7)} \\ Andrew Howroyd, Apr 18 2018

Formula

From Andrew Howroyd, Apr 18 2018: (Start)

a(n) = 5*a(n-7) - 10*a(n-14) + 10*a(n-21) - 5*a(n-28) + a(n-35).

a(7k) = 7, a(7k+1) = 2*(7*k+1), a(7k+2) = (7*k+2)*(32*k^2+38*k+9)/3, a(7k+3) = 4*(7*k+3), a(7k+4) = (7*k+4)*(8*k+6), a(7k+5) = (7*k+5)*(8*k+8)*(k+2)*(4*k+3)/3, a(7k+6) = 8*(7*k+6)*(k+1). (End)

Extensions

a(1)-a(2) and terms a(15) and beyond from Andrew Howroyd, Apr 18 2018