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A241682 Total number of unit squares appearing in the Thue-Morse sequence logical matrices after n stages.

%I A241682 #35 Sep 29 2017 02:48:45
%S A241682 0,2,0,8,16,72,240,968,3696,14792,58480,233928,932976,3731912,
%T A241682 14916720,59666888,238623856,954495432,3817806960,15271227848,
%U A241682 61084212336,244336849352,977344601200,3909378404808
%N A241682 Total number of unit squares appearing in the Thue-Morse sequence logical matrices after n stages.
%C A241682 a(n) is the total number of isolated "1s" (no adjacent 1s on horizontal and vertical) which appear as unit squares in the Thue-Morse logical matrices after n stages. See links for more details.
%H A241682 G. C. Greubel, <a href="/A241682/b241682.txt">Table of n, a(n) for n = 0..1000</a>
%H A241682 Kival Ngaokrajang, <a href="/A241682/a241682.pdf">Illustration of initial terms</a>
%H A241682 Wikipedia, <a href="http://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence">Thue-Morse sequence</a>
%H A241682 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4, 5, -20, -4, 16).
%F A241682 a(n) = A139598(A000975(n - 2)).
%F A241682 G.f.: 2*x*(12*x^4-12*x^3+x^2+4*x-1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)). - _Colin Barker_, Apr 27 2014
%F A241682 a(n) = 16*a(n-5) -4*a(n-4) -20*a(n-3) +5*a(n-2) +4*a(n-1), n>=5. (Also valid for A241683, A241684 and A241685.) - _Fung Lam_, May 02 2014
%t A241682 CoefficientList[Series[2*x*(12*x^4 - 12*x^3 + x^2 + 4*x - 1)/((x - 1)*(x + 1)*(2*x - 1)*(2*x + 1)*(4*x - 1)), {x, 0, 50}], x] (* _G. C. Greubel_, Sep 29 2017 *)
%o A241682 (PARI) {a0=0;a1=2;print1(a0,", ",a1,", "); for (n=0,50, b=ceil(2*(2^n-1)/3);a=1- (-1)^b+4*b+2*b^2; print1(a,", "))}
%o A241682 (PARI) x='x+O('x^50); concat(0, Vec(2*x*(12*x^4-12*x^3+x^2+4*x-1)/((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)))) \\ _G. C. Greubel_, Sep 29 2017
%Y A241682 Cf. A010060.
%K A241682 nonn
%O A241682 0,2
%A A241682 _Kival Ngaokrajang_, Apr 27 2014