A241893 The total number of rectangles appearing in the Thue-Morse sequence logical matrices (1, 0 version) after n stages.

0, 0, 0, 8, 28, 120, 460, 1848, 7308, 29240, 116620, 466488, 1864588, 7458360, 29827980, 119311928, 477225868, 1908903480, 7635526540, 30542106168, 122168075148, 488672300600, 1954687804300

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, 1 column or 1 row, 2 columns) that appear as rectangles 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, A241684.

Programs

Mathematica

CoefficientList[Series[4*x^3*(-2 + x + 8*x^2)/((x - 1)*(4*x - 1)*(2*x + 1)*(2*x - 1)*(1 + x)), {x, 0, 50}], x] (* G. C. Greubel, Sep 29 2017 *)

PARI

{a0=0;a=0;b=1;print1(a0,", ",a,", "); for (n=2,50, if(Mod(n,2)==0, a = 2*(a*2-(4*b-4)) + 4*b; b=b*4-2, a=a*4-8); if(Mod(n,2)==0, print1(a-4,", "),print1(a,", ")))}

Formula

a(n) = A233036(A005578(n+1)).

G.f.: 4*x^3*(-2+x+8*x^2) / ( (x-1)*(4*x-1)*(2*x+1)*(2*x-1)*(1+x) ). - R. J. Mathar, May 04 2014

a(n) = (3*2^n+2*4^n-(-1)^n*(2^n+12)-28)/18, n>0. - R. J. Mathar, May 04 2014