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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 A205248 #26 Feb 16 2025 08:33:16
%S A205248 16,40,112,328,976,2920,8752,26248,78736,236200,708592,2125768,
%T A205248 6377296,19131880,57395632,172186888,516560656,1549681960,4649045872,
%U A205248 13947137608,41841412816,125524238440,376572715312,1129718145928,3389154437776
%N 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.
%C A205248 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 A205248 R. H. Hardin, <a href="/A205248/b205248.txt">Table of n, a(n) for n = 1..210</a>
%H A205248 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ApollonianNetwork.html">Apollonian Network</a>
%H A205248 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Clique.html">Clique</a>
%H A205248 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3).
%F A205248 a(n) = 4*a(n-1) - 3*a(n-2).
%F A205248 From _Andrew Howroyd_, Sep 02 2017: (Start)
%F A205248 a(n) = 4*(3^n + 1).
%F A205248 G.f.: 8*x*(2 - 3*x)/((1 - x)*(1 - 3*x)).
%F A205248 a(n) = 8*A007051(n).
%F A205248 a(n) = 1 + A289521(n) + A067771(n) + A003462(n+1) + A003462(n).
%F A205248 (End)
%e A205248 Some solutions for n=4:
%e A205248   1  0    0  1    1  1    0  1    1  1    1  1    1  0    1  0    1  1    1  1
%e A205248   0  1    0  0    1  1    0  1    0  1    0  1    0  1    0  0    1  1    1  1
%e A205248   1  0    1  1    1  1    0  1    0  1    0  0    1  0    0  1    1  1    1  1
%e A205248   0  1    1  0    1  1    0  0    0  1    1  0    0  1    1  1    1  1    1  1
%e A205248   1  0    0  0    1  1    0  1    1  1    1  1    1  0    0  1    1  1    1  1
%t A205248 Table[4*(3^n + 1), {n, 1, 25}] (* _Jean-François Alcover_, Nov 01 2017, after _Andrew Howroyd_ *)
%t A205248 4 (3^Range[30] + 1) (* _Eric W. Weisstein_, Nov 29 2017 *)
%t A205248 LinearRecurrence[{4, -3}, {16, 40}, 30] (* _Eric W. Weisstein_, Nov 29 2017 *)
%t A205248 CoefficientList[Series[-8 (-2 + 3 x)/(1 - 4 x + 3 x^2), {x, 0, 30}], x] (* _Eric W. Weisstein_, Nov 29 2017 *)
%o A205248 (PARI) Vec(8*(2 - 3*x)/((1 - x)*(1 - 3*x)) + O(x^40)) \\ _Andrew Howroyd_, Sep 02 2017
%Y A205248 Column 1 of A205255.
%Y A205248 Cf. A003462, A007051, A067771, A289521.
%K A205248 nonn,easy
%O A205248 1,1
%A A205248 _R. H. Hardin_, Jan 24 2012