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A241685 The total number of squares and rectangles appearing in the Thue-Morse sequence logical matrices after n stages.

%I A241685 #22 Sep 29 2017 02:53:20
%S A241685 0,2,4,18,60,242,924,3698,14620,58482,233244,932978,3729180,14916722,
%T A241685 59655964,238623858,954451740,3817806962,15271053084,61084212338,
%U A241685 244336150300,977344601202,3909375608604
%N A241685 The total number of squares and rectangles appearing in the Thue-Morse sequence logical matrices after n stages.
%C A241685 a(n) is the total number of unit squares (A241682), 2 X 2 squares (A241683), 2 X 1 and 1 X 2 rectangles (A241684) that appear in the Thue-Morse logical matrices after n stages. See links for more details.
%H A241685 G. C. Greubel, <a href="/A241685/b241685.txt">Table of n, a(n) for n = 0..1000</a>
%H A241685 Kival Ngaokrajang, <a href="/A241685/a241685.pdf">Illustration of initial terms</a>
%H A241685 Kival Ngaokrajang, <a href="/A241685/a241685_1.pdf">Illustration for n = 6</a>
%H A241685 Wikipedia, <a href="http://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence">Thue-Morse sequence</a>
%H A241685 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4, 5, -20, -4, 16).
%F A241685 a(n)  = A007590(A005578(n+1)).
%F A241685 Empirical g.f.: -2*x*(4*x^3-4*x^2-2*x+1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)). - _Colin Barker_, Apr 27 2014
%F A241685 a(n) = floor((2^(n + 2) + 3 - (-1)^n)^2/72). - _G. C. Greubel_, Sep 29 2017
%t A241685 Table[Floor[(2^(n + 2) + 3 - (-1)^n)^2/72], {n, 0, 50}] (* _G. C. Greubel_, Sep 29 2017 *)
%o A241685 (PARI){for (n=1,50, b=(2^(n+1)+3+(-1)^n)/6; a=floor(b^2/2); print1(a,","))}
%Y A241685 Cf. A010060.
%K A241685 nonn
%O A241685 0,2
%A A241685 _Kival Ngaokrajang_, Apr 27 2014