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A241893 The total number of rectangles appearing in the Thue-Morse sequence logical matrices (1, 0 version) after n stages.

%I A241893 #17 Sep 29 2017 02:53:28
%S A241893 0,0,0,8,28,120,460,1848,7308,29240,116620,466488,1864588,7458360,
%T A241893 29827980,119311928,477225868,1908903480,7635526540,30542106168,
%U A241893 122168075148,488672300600,1954687804300
%N A241893 The total number of rectangles appearing in the Thue-Morse sequence logical matrices (1, 0 version) after n stages.
%C A241893 a(n) is the total number of non-isolated "1s" (consecutive 1s on 2 rows, 1 column or 1 row, 2 columns) that appear as rectangles in the Thue-Morse sequence (another version starts with 1) logical matrices after n stages. See links for more details.
%H A241893 G. C. Greubel, <a href="/A241893/b241893.txt">Table of n, a(n) for n = 0..1000</a>
%H A241893 Kival Ngaokrajang, <a href="/A241893/a241893_1.pdf">Illustration of initial terms</a>
%H A241893 Wikipedia, <a href="http://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence">Thue-Morse sequence</a>
%H A241893 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,5,-20,-4,16).
%F A241893 a(n) = A233036(A005578(n+1)).
%F A241893 G.f.: 4*x^3*(-2+x+8*x^2) / ( (x-1)*(4*x-1)*(2*x+1)*(2*x-1)*(1+x) ). - _R. J. Mathar_, May 04 2014
%F A241893 a(n) = (3*2^n+2*4^n-(-1)^n*(2^n+12)-28)/18, n>0. - _R. J. Mathar_, May 04 2014
%t A241893 CoefficientList[Series[4*x^3*(-2 + x + 8*x^2)/((x - 1)*(4*x - 1)*(2*x + 1)*(2*x - 1)*(1 + x)), {x, 0, 50}], x] (* _G. C. Greubel_, Sep 29 2017 *)
%o A241893 (PARI){a0=0;a=0;b=1;print1(a0,", ",a,", "); for (n=2,50, if(Mod(n,2)==0, a = 2*(a*2-(4*b-4)) + 4*b; b=b*4-2, a=a*4-8); if(Mod(n,2)==0, print1(a-4,", "),print1(a,", ")))}
%Y A241893 Cf. A010059, A241684.
%K A241893 nonn,easy
%O A241893 0,4
%A A241893 _Kival Ngaokrajang_, May 01 2014