OFFSET
1,3
COMMENTS
Also, the connected domination number of the n-Apollonian network. - Andrew Howroyd, Jan 16 2018
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Apollonian Network
Eric Weisstein's World of Mathematics, Connected Domination Number
Eric Weisstein's World of Mathematics, Domination Number
Index entries for linear recurrences with constant coefficients, signature (4,-3).
FORMULA
a(n) = (3^(n-3) + 5) / 2 for n > 2. - Andrew Howroyd, Sep 01 2017
From Colin Barker, Oct 03 2017: (Start)
G.f.: x*(1 - 3*x + 2*x^2 - 5*x^3) / ((1 - x)*(1 - 3*x)).
a(n) = 4*a(n-1) - 3*a(n-2) for n>4.
(End)
a(n) = A289521(n-3) for n > 3. - Andrew Howroyd, Jan 16 2018
MATHEMATICA
(* Start from Eric W. Weisstein, Jan 17 2018 *)
Join[{1, 1}, Table[(3^(n - 3) + 5)/2, {n, 3, 20}]]
Join[{1, 1}, Table[(3^n + 135)/54, {n, 3, 20}]]
Join[{1, 1}, (3^Range[3, 20] + 135)/54]
Join[{1, 1}, LinearRecurrence[{4, -3}, {3, 4}, 20]]
CoefficientList[Series[(1 - 3 x + 2 x^2 - 5 x^3)/(1 - 4 x + 3 x^2), {x, 0, 20}], x]
(* End *)
PROG
(PARI) \\ here d0..d3 are for 0..3 outside vertices included in dominating set.
D(d0, d1, d2, d3) = {[min(3*d0, 1+3*d1), min(d0+2*d1, 1+d1+2*d2), min(2*d1+d2, 1+2*d2+d3), min(3*d2, 1+3*d3)]}
a(n)={my(v=[1, 0, 0, 0]); for(i=2, n, v=D(v[1], v[2], v[3], v[4])); min(min(v[1], 1+v[2]), min(2+v[3], 3+v[4]))} \\ Andrew Howroyd, Sep 01 2017
(PARI) Vec(x*(1 - 3*x + 2*x^2 - 5*x^3) / ((1 - x)*(1 - 3*x)) + O(x^40)) \\ Colin Barker, Oct 03 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Aug 31 2017
EXTENSIONS
a(7)-a(30) from Andrew Howroyd, Sep 01 2017
STATUS
approved