A222955 Number of n X 1 0..1 arrays with every row and column least squares fitting to a zero slope straight line, with a single point array taken as having zero slope.

2, 2, 4, 4, 8, 8, 20, 18, 52, 48, 152, 138, 472, 428, 1520, 1392, 5044, 4652, 17112, 15884, 59008, 55124, 206260, 193724, 729096, 688008, 2601640, 2465134, 9358944, 8899700, 33904324, 32342236, 123580884, 118215780, 452902072, 434314138, 1667837680

Offset

1,1

Comments

Conjecture: A binary word is counted iff it has the same sum of positions of 1's as its reverse, or, equivalently, the same sum of partial sums as its reverse. - Gus Wiseman, Jan 07 2023

Links

R. H. Hardin, Table of n, a(n) for n = 1..210

Helmut Prodinger, On a generalization of the Dyck-language over a two letter alphabet, Discrete Math. 28 (1979), 269-276.

Gwenaël Richomme, On some 2-binomial coefficients of binary words: geometrical interpretation, partitions of integers, and fair words, arXiv:2510.07159 [cs.DM], 2025. See pp. 4, 28.

Formula

An asymptotic formula for a(n) can be found in Prodinger's paper.

Example

All solutions for n=4:
  ..0....1....1....0
  ..0....1....0....1
  ..0....1....0....1
  ..0....1....1....0
From Gus Wiseman, Jan 07 2023: (Start)
The a(1) = 2 through a(7) = 20 binary words with least squares fit a line of zero slope are:
  (0)  (00)  (000)  (0000)  (00000)  (000000)  (0000000)
  (1)  (11)  (010)  (0110)  (00100)  (001100)  (0001000)
             (101)  (1001)  (01010)  (010010)  (0010100)
             (111)  (1111)  (01110)  (011110)  (0011100)
                            (10001)  (100001)  (0100010)
                            (10101)  (101101)  (0101010)
                            (11011)  (110011)  (0110001)
                            (11111)  (111111)  (0110110)
                                               (0111001)
                                               (0111110)
                                               (1000001)
                                               (1000110)
                                               (1001001)
                                               (1001110)
                                               (1010101)
                                               (1011101)
                                               (1100011)
                                               (1101011)
                                               (1110111)
                                               (1111111)
(End)

Crossrefs

Column 1 of A222959.

These words appear to be ranked by A359402.

A011782 counts compositions.

A359042 adds up partial sums of standard compositions, reversed A029931.

Cf. A053632, A070925, A231204, A318283, A359043.

Sequence in context: A158780 A231208 A306663 * A217208 A287136 A351788

Adjacent sequences: A222952 A222953 A222954 * A222956 A222957 A222958

Keyword

nonn

Author

R. H. Hardin, Mar 10 2013

Status

approved