Voozh

A360286

Irregular triangle read by rows where row n is the lexicographically earliest sequence of visits, taking steps by 1, around a circle of vertices 1..n where the numbers of visits to the vertices are 1..n in some order.

1, 1, 2, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 4, 3, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 5, 4, 3, 4, 3, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 6, 5, 4, 3, 4, 3, 4, 3, 4, 5, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 7, 6, 5, 4, 3, 4, 3, 4, 3, 4, 3, 4, 5, 6, 5

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

OFFSET

1,3

COMMENTS

In each corner of an n-gon-shaped larder, a wheel of cheese is stored. The corners are numbered, going around from 1 to n. A mouse appears and takes a single bite from the cheese in one corner. Then he runs to the next corner and takes another bite from the cheese there. He carries on, running from corner to corner keeping to the wall and never crossing the floor, taking always a single bite.

He eventually reaches each corner at least once and his total consumption is one bite at one corner, two bites at some other one, three at yet another, and so on, altogether n*(n+1)/2 bites of the cheeses.

At each step the mouse has 2 choices of direction and prefers the smaller number cheese provided going that way permits a final 1..n bites (in some order).

(For n = 2 a two-cornered, curved-walled larder can be imagined, and for n = 1 a teardrop-shaped one with a single corner.)

LINKS

Table of n, a(n) for n=1..84.

EXAMPLE

Triangle begins:

k=1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

n=1: 1;

n=2: 1, 2, 1;

n=3: 1, 2, 1, 2, 1, 3;

n=4: 1, 2, 1, 2, 1, 2, 1, 4, 3, 4;

n=5: 1, 2, 1, 2, 1, 2, 1, 2, 1, 5, 4, 3, 4, 3, 4;

n=6: 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 6, 5, 4, 3, 4, 3, 4, 3, 4, 5;