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 1-2 of 2 results.

A139434 Frieze pattern with 4 rows, read by diagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2
Offset: 0

Views

Author

N. J. A. Sloane, Jun 09 2008

Keywords

Comments

Period 20: repeat [1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1]. - Wesley Ivan Hurt, Jun 05 2016
Every frieze is defined by its quiddity (the row below the row of 1's), which corresponds to the counts of triangles at vertices of a dissection of a regular polygon. The quiddity of this frieze is A135352. One can say that this frieze pattern has width 2 (not counting the rows of 1's), 4, or 5 (implying the additional row of 0's; this is also the period of the pattern and the number of vertices in the dissected polygon), depending on the convention. In any case, friezes of given width are enumerated by A000207 if we identify shifts and mirror images, otherwise by A000108. A000207(3) = 1 means that this is the only frieze of this width, and it has A000108(3) = 5 different horizontal shifts or reflections. The A000207(4) = 3 friezes having width 1 greater than this one are A139438, A139458, and one more with quiddity 1, 3, 1, 3, 1, 3, ... (currently not in the OEIS). The only frieze having width 1 less than this one has quiddity 1, 2, 1, 2, ... (A245477 can be interpreted as representing that frieze pattern). - Andrey Zabolotskiy, Jan 30 2024

Examples

			The frieze pattern is
...1 1 1 1 1 1 1 ...
....1 2 2 1 3 1 2 ...
.....1 3 1 2 2 1 3 ...
......1 1 1 1 1 1 1 ...

References

  • J. H. Conway and R. K. Guy, The Book of Numbers. New York: Springer-Verlag, p. 97, 1996.

Crossrefs

Cf. A139438, A139458, A135352, A000108, A000207, A245477.

A172051 Decimal expansion of 1/999999.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0
Offset: 0

Views

Author

Mats Granvik, Jan 24 2010

Keywords

Comments

Period 6: repeat [0, 0, 0, 0, 0, 1].

Links

Crossrefs

Cf. A022003, A172050, A245477.

Programs

  • Magma
    [n mod (1 + ((n-1) mod 3)) : n in [0..100]]; // Wesley Ivan Hurt, Aug 29 2014
  • Maple
    A172051:=n->(n mod (1+((n-1) mod 3))): seq(A172051(n), n=0..100); # Wesley Ivan Hurt, Aug 16 2014
  • Mathematica
    Join[{0,0,0,0,0},RealDigits[1/999999,10,120][[1]]] (* or *) PadRight[ {},120,{0,0,0,0,0,1}] (* Harvey P. Dale, Oct 24 2013 *)

Formula

a(n) = (1-((-1)^A172050(n+4)))/2. A similar formula is given by Hieronymus Fischer in A022003.
a(n) = 1 if (n+1) mod 6 = 0 else 0.
a(n) = A079979(n+1). [R. J. Mathar, Jan 28 2010]
a(n) = (n-2)*(Fibonacci(n-2)-1) mod 2. [Gary Detlefs, Dec 29 2010]
a(n) = n mod (1 + (n-1) mod 3). - Wesley Ivan Hurt, Aug 16 2014
G.f.: x^5/(1-x^6). - Vaclav Kotesovec, Aug 18 2014
a(n) = binomial((5*n+10) mod 6, 5). - Wesley Ivan Hurt, Aug 29 2014
a(n) = 1 - sign((n+1) mod 6). - Wesley Ivan Hurt, Aug 29 2014
a(n) = A245477(n) - 1. - Wesley Ivan Hurt, Aug 29 2014
From Wesley Ivan Hurt, Jun 23 2016: (Start)
a(n) = a(n-6) for n>5.
a(n) = (3 - 3*cos(n*Pi) + 6*cos(n*Pi/3) + 6*cos((n-4)*Pi/3) - 2*sqrt(3)*sin(2*n*Pi/3) - 2*sqrt(3)*sin((1+2*n)*Pi/3))/18. (End)
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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