cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Kival Ngaokrajang, Apr 27 2014

Keywords

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.

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

A241685 The total number of squares and rectangles appearing in the Thue-Morse sequence logical matrices after n stages.

Original entry on oeis.org

0, 2, 4, 18, 60, 242, 924, 3698, 14620, 58482, 233244, 932978, 3729180, 14916722, 59655964, 238623858, 954451740, 3817806962, 15271053084, 61084212338, 244336150300, 977344601202, 3909375608604
Offset: 0

Views

Author

Kival Ngaokrajang, Apr 27 2014

Keywords

Comments

a(n) is the total number of unit squares (A241682), 2 X 2 squares (A241683), 2 X 1 and 1 X 2 rectangles (A241684) that appear in the Thue-Morse logical matrices after n stages. See links for more details.

Crossrefs

Cf. A010060.

Programs

  • Mathematica
    Table[Floor[(2^(n + 2) + 3 - (-1)^n)^2/72], {n, 0, 50}] (* G. C. Greubel, Sep 29 2017 *)
  • PARI
    {for (n=1,50, b=(2^(n+1)+3+(-1)^n)/6; a=floor(b^2/2); print1(a,","))}

Formula

a(n) = A007590(A005578(n+1)).
Empirical g.f.: -2*x*(4*x^3-4*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) = floor((2^(n + 2) + 3 - (-1)^n)^2/72). - G. C. Greubel, Sep 29 2017

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

Original entry on oeis.org

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

Views

Author

Kival Ngaokrajang, May 01 2014

Keywords

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.

Crossrefs

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
Showing 1-3 of 3 results.