A241682 -id:A241682 - cp's OEIS frontend

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

0, 2, 4, 18, 60, 242, 924, 3698, 14620, 58482, 233244, 932978, 3729180, 14916722, 59655964, 238623858, 954451740, 3817806962, 15271053084, 61084212338, 244336150300, 977344601202, 3909375608604

Offset: 0

Kival Ngaokrajang, Apr 27 2014

nonn

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.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Kival Ngaokrajang, Illustration for n = 6
Wikipedia, Thue-Morse sequence
Index entries for linear recurrences with constant coefficients, signature (4, 5, -20, -4, 16).

Cf. A010060.

Table[Floor[(2^(n + 2) + 3 - (-1)^n)^2/72], {n, 0, 50}] (* G. C. Greubel, Sep 29 2017 *)

{for (n=1,50, b=(2^(n+1)+3+(-1)^n)/6; a=floor(b^2/2); print1(a,","))}

a(n)  = A007590(A005578(n+1)).
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
a(n) = floor((2^(n + 2) + 3 - (-1)^n)^2/72). - G. C. Greubel, Sep 29 2017

A241891 Total number of unit squares appearing in the Thue-Morse sequence of logical matrices (1, 0 version) after n stages.

Original entry on oeis.org

1, 2, 4, 8, 20, 72, 244, 968, 3700, 14792, 58484, 233928, 932980, 3731912, 14916724, 59666888, 238623860, 954495432, 3817806964, 15271227848, 61084212340, 244336849352, 977344601204, 3909378404808, 15637502434420

Offset: 0

Kival Ngaokrajang, May 01 2014

a(n) is the total number of isolated "1s" (no adjacent 1s in the horizontal or vertical directions) which appear as unit squares in the Thue-Morse sequence (another version starts with 1) of logical matrices after n stages. See links for more details.

Colin Barker, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Illustration of initial terms
Wikipedia, Thue-Morse sequence
Index entries for linear recurrences with constant coefficients, signature (4,5,-20,-4,16).

Cf. A010059, A241682.

{a0=1;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; if(Mod(n,2)==0, a=a+4); print1(a,", "))}

Vec(-(24*x^5+12*x^4+2*x^3-9*x^2-2*x+1)/((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)) + O(x^100)) \\ Colin Barker, Jan 17 2015

a(n) = A033691(A001045(n)) for n > 2, a(0) = 1, a(1) = 2, a(2) = 4.
a(n) = 4*a(n-1)+5*a(n-2)-20*a(n-3)-4*a(n-4)+16*a(n-5). - Colin Barker, Jan 17 2015
G.f.: -(24*x^5+12*x^4+2*x^3-9*x^2-2*x+1) / ((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)). - Colin Barker, Jan 17 2015

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

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A241891 Total number of unit squares appearing in the Thue-Morse sequence of logical matrices (1, 0 version) after n stages.

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PARI

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