A205248 - OEIS

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.

%I #26 Feb 16 2025 08:33:16

%S 16,40,112,328,976,2920,8752,26248,78736,236200,708592,2125768,

%T 6377296,19131880,57395632,172186888,516560656,1549681960,4649045872,

%U 13947137608,41841412816,125524238440,376572715312,1129718145928,3389154437776

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

%C 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

%H R. H. Hardin, <a href="/A205248/b205248.txt">Table of n, a(n) for n = 1..210</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ApollonianNetwork.html">Apollonian Network</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Clique.html">Clique</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3).

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

%F From _Andrew Howroyd_, Sep 02 2017: (Start)

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

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

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

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

%F (End)

%e Some solutions for n=4:

%e 1 0 0 1 1 1 0 1 1 1 1 1 1 0 1 0 1 1 1 1

%e 0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 1 1 1 1

%e 1 0 1 1 1 1 0 1 0 1 0 0 1 0 0 1 1 1 1 1

%e 0 1 1 0 1 1 0 0 0 1 1 0 0 1 1 1 1 1 1 1

%e 1 0 0 0 1 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1

%t Table[4*(3^n + 1), {n, 1, 25}] (* _Jean-François Alcover_, Nov 01 2017, after _Andrew Howroyd_ *)

%t 4 (3^Range[30] + 1) (* _Eric W. Weisstein_, Nov 29 2017 *)

%t LinearRecurrence[{4, -3}, {16, 40}, 30] (* _Eric W. Weisstein_, Nov 29 2017 *)

%t CoefficientList[Series[-8 (-2 + 3 x)/(1 - 4 x + 3 x^2), {x, 0, 30}], x] (* _Eric W. Weisstein_, Nov 29 2017 *)

%o (PARI) Vec(8*(2 - 3*x)/((1 - x)*(1 - 3*x)) + O(x^40)) \\ _Andrew Howroyd_, Sep 02 2017

%Y Column 1 of A205255.

%Y Cf. A003462, A007051, A067771, A289521.

%K nonn,easy,changed

%O 1,1

%A _R. H. Hardin_, Jan 24 2012