A241684 The total number of rectangles appearing in the Thue-Morse sequence logical matrices after n stages.

0, 0, 4, 8, 32, 120, 464, 1848, 7312, 29240, 116624, 466488, 1864592, 7458360, 29827984, 119311928, 477225872, 1908903480, 7635526544, 30542106168, 122168075152, 488672300600, 1954687804304, 7818751217208, 31274999276432, 125099997105720, 500399966053264, 2001599864213048

Offset: 0

Author

Kival Ngaokrajang, Apr 27 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 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

Magma

[(8+3*2^n+2*4^n+(-1)^n*(24-2^n))/18: n in [0..30]]; // Vincenzo Librandi, Sep 29 2017

Mathematica

CoefficientList[Series[-4*x^2*(8*x^3 - 5*x^2 - 2*x + 1)/((x - 1)*(x + 1)*(2*x - 1)*(2*x + 1)*(4*x - 1)), {x, 0, 50}], x] (* G. C. Greubel, Sep 28 2017 *)

PARI

x='x+O('x^50); concat([0,0], Vec(-4*x^2*(8*x^3-5*x^2-2*x+1)/((x-1)*(x+1)*(2*x-1)*(2*x+1)*(4*x-1)))) \\ G. C. Greubel, Sep 28 2017

Formula

a(n) = A007590(A005578(n+1)) - (A139598(A000975(n-2)) + A007590(A000975(n-1))).

G.f.: -4*x^2*(8*x^3-5*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) = (8 + 3*2^n + 2*4^n + (-1)^n*(24 - 2^n))/18, n>0. - R. J. Mathar, May 04 2014

Extensions

Terms a(21) onward added by G. C. Greubel, Sep 28 2017