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

1,1

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, 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).

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)

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]

Prog

(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

Crossrefs

Cf. A302255, A302760, A302763.

Sequence in context: A335327 A163895 A309016 * A180071 A034882 A175943

Adjacent sequences: A302649 A302650 A302651 * A302653 A302654 A302655

Keyword

nonn,easy

Author

Eric W. Weisstein, Apr 11 2018

Extensions

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

Status

approved