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

Author

Kival Ngaokrajang, Apr 27 2014

Keywords

• nonn

Comments

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.

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. A010060.

Programs

Mathematica

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 *)

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,", "))}

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

Formula

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