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A241894 The total number of squares and rectangles appearing in the Thue-Morse sequence (1, 0 version) logical matrices after n stages.

%I A241894 #16 Aug 02 2016 10:16:45
%S A241894 1,2,5,18,61,242,925,3698,14621,58482,233245,932978,3729181,14916722,
%T A241894 59655965,238623858,954451741,3817806962,15271053085,61084212338,
%U A241894 244336150301,977344601202,3909375608605
%N A241894 The total number of squares and rectangles appearing in the Thue-Morse sequence (1, 0 version) logical matrices after n stages.
%C A241894 a(n) is the total number of unit squares (A241891), 2 X 2
%C A241894 squares (A241892), 2 X 1 and 1 X 2 rectangles (A241893) that appear in the Thue-Morse sequence (another version starts with 1) logical matrices after n stages. See links for more details.
%H A241894 Kival Ngaokrajang, <a href="/A241894/a241894_2.pdf">Illustration of initial terms</a>
%H A241894 Kival Ngaokrajang, <a href="/A241894/a241894_1.pdf">illustration for n = 6</a>
%H A241894 Wikipedia, <a href="http://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence">Thue-Morse sequence</a>
%H A241894 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,5,-20,-4,16).
%F A241894 a(n) = A000982(A005578(n+1)).
%F A241894 G.f.: ( -1+2*x+8*x^2-8*x^3-8*x^4 ) / ( (x-1)*(4*x-1)*(1+2*x)*(2*x-1)*(1+x) ). - _R. J. Mathar_, May 04 2014
%F A241894 18*a(n) = 7+6*2^n +4^(n+1) +(-1)^n*( 3-2^(n+1) ). - _R. J. Mathar_, May 04 2014
%t A241894 LinearRecurrence[{4,5,-20,-4,16},{1,2,5,18,61},30] (* _Harvey P. Dale_, Aug 02 2016 *)
%o A241894 (PARI){a0=1; print1(a0,", "); for (n=2,50, b=(2^(n+1)+3+(-1)^n)/6; a=floor(b^2/2);if(Mod(n,2)==1, a=a+1); print1(a,", "))}
%Y A241894 Cf. A010059, A241685.
%K A241894 nonn,easy
%O A241894 0,2
%A A241894 _Kival Ngaokrajang_, May 01 2014