A385862 -id:A385862 - cp's OEIS frontend

A384764 Number of uniquely solveable n X m nonograms (hanjie), read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 14, 8, 1, 1, 16, 52, 52, 16, 1, 1, 32, 210, 384, 210, 32, 1, 1, 64, 816, 3152, 3152, 816, 64, 1, 1, 128, 3206, 24230, 52362, 24230, 3206, 128, 1, 1, 256, 12536, 189898, 814632, 814632, 189898, 12536, 256, 1, 1, 512, 48962, 1473674, 12819322, 25309575, 12819322, 1473674, 48962, 512, 1

Offset: 0

Bertram Felgenhauer, Jun 09 2025

In this game there is an n X m grid where each square may or may not be filled. Each column and each row is labeled by the length of each successive block of filled squares, but without indication of the number of unfilled squares in between. The object is to determine which squares are filled.

			A(2,2) = 16-2 because out of the possible 2^(2*2) grids, only 10/01 and 01/10 have the same row and column clues.
Top left corner of the array:
  1,  1,    1,      1,        1,         1,           1, ...
  1,  2,    4,      8,       16,        32,          64, ...
  1,  4,   14,     52,      210,       816,        3206, ...
  1,  8,   52,    384,     3152,     24230,      189898, ...
  1, 16,  210,   3152,    52362,    814632,    12819322, ...
  1, 32,  816,  24230,   814632,  25309575,   794378773, ...
  1, 64, 3206, 189898, 12819322, 794378773, 49745060669, ...

Bertram Felgenhauer, Antidiagonals n+m = 0..13, flattened
Bertram Felgenhauer, Counting Nonograms.
Wikipedia, Nonogram.

Cf. A242876 (main diagonal), A000012 (column m=0), A000079 (column m=1), A383345 (column m=2).

Cf. A385862 (variant: uniquely solveable n X m yesnograms).

Basic properties include A(n,m) = A(m,n), A(n,m) <= 2^(n*m), A(0,n) = A(n,0) = 1, and A(1,n) = A(n,1) = 2^n.

A383345 Number of uniquely solveable n X 2 nonograms (hanjie).

Original entry on oeis.org

1, 4, 14, 52, 210, 816, 3206, 12536, 48962, 191226, 746456, 2913544, 11371040, 44376798, 173181564, 675834086, 2637392942, 10292179494, 40164144690, 156736057740, 611644171812, 2386868430698, 9314465669046

Offset: 0

Bertram Felgenhauer, Jun 11 2025

In this game there is an n X 2 grid where each square may or may not be filled. Each column and each row is labeled by the length of each successive block of filled squares, but without indication of the number of unfilled squares in between. The object is to determine which squares are filled.

The only ambiguous row hint is 1, which has the same solutions regardless of whether black or white squares are counted. So this is also the number of n X 2 "yesnograms".

			a(2) = 16-2 because out of the possible 2^(2*2) grids, only 10/01 and 01/10 have the same row and column clues.

Bertram Felgenhauer, Counting Nonograms.
Wikipedia, Nonogram.

Column m=2 of A384764. Also column m=2 of A385862 (n X m yesnograms).

Cf. A242876.

A385861 Number of n X n yesnograms that can be solved uniquely.

Original entry on oeis.org

1, 2, 14, 368, 49578, 24177516, 46985524156

Offset: 0

Karl W. Heuer, Aug 06 2025

A nonogram provides row and column clues indicating runs of black pixels, treating white as blank. In this variant (called a "yesnogram"), the row clues instead indicate runs of white pixels, treating black as blank. Column clues remain unchanged from the standard nonogram.

			a(2) = 14 because, of the 16 2 X 2 grids, 10/01 and 01/10 would have the same set of clues; the other 14 are solvable.

Bertram Felgenhauer, Counting Nonograms.
Wikipedia, Nonogram.

Main diagonal of A385862.

Cf. A242876 (solvable n X n nonograms), A384764 (solvable n X m nonograms), A383345 (solvable n X 2 nonograms or yesnograms).

cp's OEIS Frontend

A384764 Number of uniquely solveable n X m nonograms (hanjie), read by antidiagonals.

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A383345 Number of uniquely solveable n X 2 nonograms (hanjie).

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A385861 Number of n X n yesnograms that can be solved uniquely.

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