A337515 a(n) is the number of essentially different optimal solutions with A336660(n) 1 X 1 cells to R. Kurchan's n-squares puzzle 578.

1, 1, 1, 4, 1, 2, 1, 8, 4, 21, 29

Offset

1,4

Comments

Numbers a(n) are a by-product of the exhaustive computation done for A336660. They were communicated privately by Hugo Pfoertner and Hermann Jurksch.

For historical background see link section "Pictures of all different solutions for n <= 11".

Links

Table of n, a(n) for n=1..11.

Rainer Rosenthal, Illustration of a(4).

Rainer Rosenthal, Illustration of a(8).

Rainer Rosenthal, Data of all different solutions for n <= 11, October 2020.

Rainer Rosenthal, Pictures of all different solutions for n <= 11, October 2020.

Example

The square with side length s has lower left corner (x_s,y_s).
The a(8) = 8 solutions come in 6 main variations:
       x8 y8  x7 y7  x6 y6  x5 y5  x4 y4  x3 y3  x2 y2  x1 y1
  V1    0  0   0  2   4  6   5  5   2  7   6  4   1  7   4 10
  V2    0  0   2  3   7  0   5  2   7  5   6  1   8  8  10  7
  V3+   0  0   3  7   0  3   2  6   5  5   1  7   7  4   3  6
    +  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   8  8
  V4+   0  0   3  7   0  3   2  6   5  6   1  7   6  9   4 10
    +  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..  .. ..   8  8
  V5    0  0   4  6   1  3   5  5   2  7   6  4   3 10   2  9
  V6    0  0   4  6   5  5   2  4   6  3   3  7   7  2   9  4
The two solutions in V3 are "1-siblings", i.e., they are equal except for (x1,y1).
Likewise V4 consists of two 1-siblings. See illustration of a(8) in the link section.

Crossrefs

Cf. A336659, A336660, A336782.

Sequence in context: A126241 A353515 A019777 * A090885 A008476 A300657

Adjacent sequences: A337512 A337513 A337514 * A337516 A337517 A337518

Keyword

nonn,hard,more

Author

Rainer Rosenthal, Sep 19 2020

Extensions

a(4) and a(8) corrected by Rainer Rosenthal, Oct 02 2020

Status

approved