A241685 -id:A241685 - cp's OEIS frontend

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

0, 2, 0, 8, 16, 72, 240, 968, 3696, 14792, 58480, 233928, 932976, 3731912, 14916720, 59666888, 238623856, 954495432, 3817806960, 15271227848, 61084212336, 244336849352, 977344601200, 3909378404808

Offset: 0

Kival Ngaokrajang, Apr 27 2014

nonn

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.

G. C. Greubel, 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. A010060.

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 *)

{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,", "))}

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

a(n) = A139598(A000975(n - 2)).
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
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

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

Original entry on oeis.org

1, 2, 5, 18, 61, 242, 925, 3698, 14621, 58482, 233245, 932978, 3729181, 14916722, 59655965, 238623858, 954451741, 3817806962, 15271053085, 61084212338, 244336150301, 977344601202, 3909375608605

Offset: 0

Kival Ngaokrajang, May 01 2014

a(n) is the total number of unit squares (A241891), 2 X 2

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.

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. A010059, A241685.

LinearRecurrence[{4,5,-20,-4,16},{1,2,5,18,61},30] (* Harvey P. Dale, Aug 02 2016 *)

{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,", "))}

a(n) = A000982(A005578(n+1)).
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
18*a(n) = 7+6*2^n +4^(n+1) +(-1)^n*( 3-2^(n+1) ). - R. J. Mathar, May 04 2014

cp's OEIS Frontend

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

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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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula