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A302652
Number of minimum total dominating sets in the n-antiprism graph.
4
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
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
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