A139434 - OEIS

A139434

Frieze pattern with 4 rows, read by diagonals.

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

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

REFERENCES

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

LINKS

Table of n, a(n) for n=0..98.

EXAMPLE

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 ...

CROSSREFS

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