A233036 -id:A233036 - cp's OEIS frontend

A233035 a(n) = n * floor(n/4).

0, 0, 0, 4, 5, 6, 7, 16, 18, 20, 22, 36, 39, 42, 45, 64, 68, 72, 76, 100, 105, 110, 115, 144, 150, 156, 162, 196, 203, 210, 217, 256, 264, 272, 280, 324, 333, 342, 351, 400, 410, 420, 430, 484, 495, 506, 517, 576, 588, 600, 612, 676, 689, 702, 715, 784, 798, 812, 826, 900, 915, 930, 945, 1024, 1040

Offset: 1

Kival Ngaokrajang, Dec 03 2013

The maximum number of I patterns tetrominos that can be packed into an n X n array of squares with rotation is prohibited.
u(n) = n*(n mod 4), where u(n) is total number of squares left after packing I patterns into n X n squares.
a(n) = A132028(n) for 4 <= n <= 31.

Kival Ngaokrajang, Illustration of initial terms
Wikipedia, Tetromino
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Cf. A132028, A233036, A242669.

Table[n*Floor[n/4],{n,80}] (* or *) LinearRecurrence[{1,0,0,2,-2,0,0,-1,1},{0,0,0,4,5,6,7,16,18},80] (* Harvey P. Dale, Aug 22 2020 *)

a(n) = n * floor(n/4); \\ Joerg Arndt, Dec 08 2013

a(n) = (n^2 - n*(n mod 4))/4.

G.f.: (x^7 + x^6 + x^5 + x^4 + 4*x^3)/((1-x)*(1-x^4)^2). - Ralf Stephan, Dec 08 2013

A233735 G.f.: x^3(x^21 - x^20 - x^11 + x^10 + x^9 - x^8 + x^6 - x^5 + x^3 + x^2 - x + 1) / ((1-x^5) (1-x)^2).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 6, 8, 10, 13, 16, 20, 25, 29, 34, 39, 45, 52, 58, 65, 72, 80, 88, 96, 105, 114, 124, 134, 144, 155, 166, 178, 190, 202, 215, 228, 242, 256, 270, 285, 300, 316, 332, 348, 365, 382, 400, 418, 436, 455, 474, 494, 514, 534

Offset: 0

Kival Ngaokrajang, Dec 15 2013

The second differences repeat with period 1,0,1,0,0 for n >= 20.

a(n) is a lower bound on A085577(n-2). The Ngaokrajang link shows arrangements of a(n) Greek crosses in an n X n grid. Note that a(11)=16, whereas A085577(9)=17, so the bound is not always tight. - N. J. A. Sloane, Apr 19 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kival Ngaokrajang, Packings of a(n) Greek crosses. [Note that it is possible to pack 17 Greek crosses into an 11 X 11 grid (see A085577), so these arrangements are not always optimal. - _N. J. A. Sloane_, Apr 19 2015]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1)

Cf. A085576, A085577, A233035, A233036.

CoefficientList[Series[x^3*(x^21 - x^20 - x^11 + x^10 + x^9 - x^8 + x^6 - x^5 + x^3 +x^2 - x + 1)/((1 - x^5)*(1 - x)^2), {x, 0, 50}], x] (* G. C. Greubel, Jan 08 2018 *)

x='x+O('x^50); Vec(x^3*(x^21 - x^20 - x^11 + x^10 + x^9 - x^8 + x^6 - x^5 + x^3 +x^2 - x + 1)/((1 - x^5)*(1 - x)^2)) \\ G. C. Greubel, Jan 08 2018

Entry revised by N. J. A. Sloane, Apr 19 2015. The new definition is a g.f. found by Ralf Stephan on Dec 17 2013. The old definition was wrong.

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

Kival Ngaokrajang, May 01 2014

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.

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).

Cf. A010059, A241684.

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 *)

{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,", ")))}

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

cp's OEIS Frontend

A233035 a(n) = n * floor(n/4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A233735 G.f.: x^3(x^21 - x^20 - x^11 + x^10 + x^9 - x^8 + x^6 - x^5 + x^3 + x^2 - x + 1) / ((1-x^5) (1-x)^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

cp's OEIS Frontend

A233035 a(n) = n * floor(n/4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A233735 G.f.: x^3*(x^21 - x^20 - x^11 + x^10 + x^9 - x^8 + x^6 - x^5 + x^3 + x^2 - x + 1) / ((1-x^5) * (1-x)^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A233735 G.f.: x^3(x^21 - x^20 - x^11 + x^10 + x^9 - x^8 + x^6 - x^5 + x^3 + x^2 - x + 1) / ((1-x^5) (1-x)^2).