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

Author

R. H. Hardin, Jan 24 2012

Keywords

• nonn
• easy

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

Examples

			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
		

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).

Crossrefs

Column 1 of A205255.

Cf. A003462, A007051, A067771, A289521.

Programs

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 *)

PARI

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

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)