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%I #20 Nov 05 2022 19:46:44
%S 0,0,1,2,13,50,221,882,3613,14450,58141,232562,931613,3726450,
%T 14911261,59645042,238602013,954408050,3817719581,15270878322,
%U 61083862813,244335451250,977343203101,3909372812402,15637496842013
%N Total number of 2 X 2 squares appearing in the Thue-Morse sequence logical matrices (1, 0 version) after n stages.
%C a(n) is the total number of non-isolated "1s" (consecutive 1s on 2 rows, 2 columns) that appear as 2 X 2 squares in the Thue-Morse sequence (another version starts with 1) logical matrices after n stages. See links for more details.
%H G. C. Greubel, <a href="/A241892/b241892.txt">Table of n, a(n) for n = 0..1000</a>
%H Kival Ngaokrajang, <a href="/A241892/a241892_1.pdf">Illustration of initial terms</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence">Thue-Morse sequence</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,5,-20,-4,16).
%F a(n) = A000982(A000975(n-1)) for n > 0, a(0) = 0.
%F G.f.: -x^2*(1-2*x+8*x^3) / ( (x-1)*(4*x-1)*(2*x+1)*(2*x-1)*(1+x) ). - _R. J. Mathar_, May 04 2014
%F 18*a(n) = 4^n+7 -3*2^n +(-1)^n*(3+2^n), n>0. - _R. J. Mathar_, May 04 2014
%t CoefficientList[Series[-x^2*(1 - 2*x + 8*x^3)/((x - 1)*(4*x - 1)*(2*x + 1)*(2*x - 1)*(1 + x)), {x, 0, 50}], x] (* _G. C. Greubel_, Oct 11 2017 *)
%t LinearRecurrence[{4, 5, -20, -4, 16}, {0, 0, 1, 2, 13, 50}, 30] (* _Harvey P. Dale_, Nov 05 2022 *)
%o (PARI){a0=0;print1(a0,", "); for (n=0,50, b=ceil(2*(2^n-1)/3); a=floor(b^2/2); if(Mod(n,2)==1, a=a+1); print1(a,", "))}
%Y Cf. A010059, A241683.
%K nonn,easy
%O 0,4
%A _Kival Ngaokrajang_, May 01 2014
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