OFFSET
0,1
COMMENTS
Sequence extended to n=0 using the recurrence. - Andrew Howroyd, May 23 2025
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Dominating Set.
Eric Weisstein's World of Mathematics, Trapezohedral Graph.
Index entries for linear recurrences with constant coefficients, signature (12,-50,75,5,-66,-12,11,18,8).
FORMULA
G.f.: (2 - 15*x + 37*x^2 - 57*x^3 + 62*x^4 + 135*x^5 - 111*x^6 - 119*x^7 - 30*x^8)/((1 - x)*(1 - 4*x)*(1 - x - x^2)*(1 - 3*x - 2*x^2)*(1 - 3*x - x^2 - x^3)). - Andrew Howroyd, May 23 2025
a(n) = 12*a(n-1)-50*a(n-2)+75*a(n-3)+5*a(n-4)-66*a(n-5)-12*a(n-6)+11*a(n-7)+18*a(n-8)+8*a(n-9). - Eric W. Weisstein, Aug 29 2025
MATHEMATICA
Table[4^n - 2 - 2 LucasL[n] + ((3 - Sqrt[17])^n + (3 + Sqrt[17])^n)/2^(n - 1) + RootSum[-1 - # - 3 #^2 + #^3 &, #^n &], {n, 0, 20}] // Expand (* Eric W. Weisstein, Aug 29 2025 *)
LinearRecurrence[{12, -50, 75, 5, -66, -12, 11, 18, 8}, {9, 45, 183, 693, 2589, 9639, 35933, 134373, 504435}, {0, 20}]
CoefficientList[Series[(-2 + 15 x - 37 x^2 + 57 x^3 - 62 x^4 - 135 x^5 + 111 x^6 + 119 x^7 + 30 x^8)/((-1 + x) (-1 + 4 x) (-1 + x + x^2) (-1 + 3 x + 2 x^2) (-1 + 3 x + x^2 + x^3)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 29 2025 *)
PROG
(PARI) Vec((2 - 15*x + 37*x^2 - 57*x^3 + 62*x^4 + 135*x^5 - 111*x^6 - 119*x^7 - 30*x^8)/((1 - x)*(1 - 4*x)*(1 - x - x^2)*(1 - 3*x - 2*x^2)*(1 - 3*x - x^2 - x^3)) + O(x^30)) \\ Andrew Howroyd, May 23 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Feb 09 2024
EXTENSIONS
a(0)-a(2) and a(15) onwards from Andrew Howroyd, May 23 2025
STATUS
approved