OFFSET
1,1
COMMENTS
Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, Apr 16 2018
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
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 (3, 0, 4, -2, 10, 4, 0, -1, -1).
FORMULA
From Andrew Howroyd, Apr 16 2018: (Start)
G.f.: x*(3 + 12*x^2 - 8*x^3 + 50*x^4 + 24*x^5 - 8*x^7 - 9*x^8)/((1 - x + x^2 + x^3)*(1 + x + x^2 - x^3)*(1 - 3*x - x^2 - x^3)).
a(n) = 3*a(n-1) + 4*a(n-3) - 2*a(n-4) + 10*a(n-5) + 4*a(n-6) - a(n-8) - a(n-9) for n > 9. (End)
MATHEMATICA
Table[RootSum[-1 - # - 3 #^2 + #^3 &, #^n &] + RootSum[1 + # - #^2 + #^3 &, #^n &] + RootSum[-1 + # + #^2 + #^3 &, #^n &], {n, 20}]
LinearRecurrence[{3, 0, 4, -2, 10, 4, 0, -1, -1}, {3, 9, 39, 121, 443,
1521, 5071, 17161, 58035}, 20]
CoefficientList[Series[(3 + 12 x^2 - 8 x^3 + 50 x^4 + 24 x^5 - 8 x^7 - 9 x^8)/(1 - 3 x - 4 x^3 + 2 x^4 - 10 x^5 - 4 x^6 + x^8 + x^9), {x, 0, 20}], x]
PROG
(PARI) Vec((3 + 12*x^2 - 8*x^3 + 50*x^4 + 24*x^5 - 8*x^7 - 9*x^8)/((1 - x + x^2 + x^3)*(1 + x + x^2 - x^3)*(1 - 3*x - x^2 - x^3)) + O(x^30)) \\ Andrew Howroyd, Apr 16 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Apr 16 2018
EXTENSIONS
a(1)-a(2) and terms a(10) and beyond from Andrew Howroyd, Apr 16 2018
STATUS
approved