A205248 Number of (n+1) X 2 0..1 arrays with the number of clockwise edge increases in every 2 X 2 subblock the same.

16, 40, 112, 328, 976, 2920, 8752, 26248, 78736, 236200, 708592, 2125768, 6377296, 19131880, 57395632, 172186888, 516560656, 1549681960, 4649045872, 13947137608, 41841412816, 125524238440, 376572715312, 1129718145928, 3389154437776

Offset

1,1

Comments

Also, the number of cliques in the n-Apollonian network. Cliques in this graph have a maximum size of 4. - Andrew Howroyd, Sep 02 2017

Links

R. H. Hardin, Table of n, a(n) for n = 1..210

Eric Weisstein's World of Mathematics, Apollonian Network

Eric Weisstein's World of Mathematics, Clique

Index entries for linear recurrences with constant coefficients, signature (4,-3).

Formula

a(n) = 4*a(n-1) - 3*a(n-2).

From Andrew Howroyd, Sep 02 2017: (Start)

a(n) = 4*(3^n + 1).

G.f.: 8*x*(2 - 3*x)/((1 - x)*(1 - 3*x)).

a(n) = 8*A007051(n).

a(n) = 1 + A289521(n) + A067771(n) + A003462(n+1) + A003462(n).

(End)

Example

Some solutions for n=4:
  1  0    0  1    1  1    0  1    1  1    1  1    1  0    1  0    1  1    1  1
  0  1    0  0    1  1    0  1    0  1    0  1    0  1    0  0    1  1    1  1
  1  0    1  1    1  1    0  1    0  1    0  0    1  0    0  1    1  1    1  1
  0  1    1  0    1  1    0  0    0  1    1  0    0  1    1  1    1  1    1  1
  1  0    0  0    1  1    0  1    1  1    1  1    1  0    0  1    1  1    1  1

Mathematica

Table[4*(3^n + 1), {n, 1, 25}] (* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)
4 (3^Range[30] + 1) (* Eric W. Weisstein, Nov 29 2017 *)
LinearRecurrence[{4, -3}, {16, 40}, 30] (* Eric W. Weisstein, Nov 29 2017 *)
CoefficientList[Series[-8 (-2 + 3 x)/(1 - 4 x + 3 x^2), {x, 0, 30}], x] (* Eric W. Weisstein, Nov 29 2017 *)

Prog

(PARI) Vec(8*(2 - 3*x)/((1 - x)*(1 - 3*x)) + O(x^40)) \\ Andrew Howroyd, Sep 02 2017

Crossrefs

Column 1 of A205255.

Cf. A003462, A007051, A067771, A289521.

Sequence in context: A368078 A185790 A185761 * A205186 A197903 A300901

Adjacent sequences: A205245 A205246 A205247 * A205249 A205250 A205251

Keyword

nonn,easy

Author

R. H. Hardin, Jan 24 2012

Status

approved