A241892 Total number of 2 X 2 squares appearing in the Thue-Morse sequence logical matrices (1, 0 version) after n stages.

0, 0, 1, 2, 13, 50, 221, 882, 3613, 14450, 58141, 232562, 931613, 3726450, 14911261, 59645042, 238602013, 954408050, 3817719581, 15270878322, 61083862813, 244335451250, 977343203101, 3909372812402, 15637496842013

Offset: 0

Author

Kival Ngaokrajang, May 01 2014

Keywords

• nonn
• easy

Comments

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.

Links

• 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).

Crossrefs

Cf. A010059, A241683.

Programs

Mathematica

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 *)
LinearRecurrence[{4, 5, -20, -4, 16}, {0, 0, 1, 2, 13, 50}, 30] (* Harvey P. Dale, Nov 05 2022 *)

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,", "))}

Formula

a(n) = A000982(A000975(n-1)) for n > 0, a(0) = 0.

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

18*a(n) = 4^n+7 -3*2^n +(-1)^n*(3+2^n), n>0. - R. J. Mathar, May 04 2014