A355552 -id:A355552 - cp's OEIS frontend

A355553 Number of ways to select 3 or more collinear points from an n X n grid.

0, 0, 8, 54, 228, 708, 1980, 4890, 11528, 26004, 57384, 123786, 265596, 563664, 1192220, 2511474, 5279208, 11064216, 23156448, 48361110, 100859180, 209996772, 436635396, 906562842, 1879950384, 3893566872, 8054935784, 16645591974, 34363631412, 70872295524, 146036933100

Offset: 1

Thomas Garrison, Jul 14 2022

nonn

			a(4)=54: There are 4 horizontal lines of length 4 and within a line of 4 dots are 5 ways to select a line 3 or longer. There are 2 diagonal lines of length 4 and 4 vertical lines of length 4. Finally there are 4 diagonals of length 3 these are: ((1,2),(2,3),(3,4)), ((2,1),(3,2),(4,3)), ((1,3),(2,2),(3,1)), ((2,4),(3,3),(4,2)). In total we have 5*10+4=54.
    4 . . . .
    3 . . . .
    2 . . . .
    1 . . . .
      1 2 3 4

Lucas A. Brown, Table of n, a(n) for n = 1..71
Lucas A. Brown, A355553.py.

Cf. A000982 (1 X n), 2*A000982 (2 X n), A355551 (3 X n), A355552 (4 X n).

Corrected and extended by Lucas A. Brown, Nov 06 2022

A355551 Number of ways to select 3 or more collinear points from a 3 X n grid.

Original entry on oeis.org

1, 2, 8, 23, 61, 144, 322, 689, 1439, 2954, 6004, 12123, 24385, 48932, 98054, 196325, 392899, 786078, 1572472, 3145295, 6290981, 12582392, 25165258, 50331033, 100662631, 201325874, 402652412, 805305539, 1610611849, 3221224524, 6442449934

Offset: 1

Thomas Garrison, Jul 06 2022

			a(5)=61: there are 3*(2^5 - 1 - binomial(6,2)) ways to select 3 or more points on a horizontal line, 5 ways on a vertical line, 3 ways on a diagonal line with slope 1, 3 ways on a diagonal line with slope -1, 1 way on a diagonal line with slope 1/2, and 1 way on a diagonal line with slope -1/2; 48 + 5 + 6 + 2 = 61.

Thomas Garrison, Table of n, a(n) for n = 1...1000
Index entries for linear recurrences with constant coefficients, signature (4,-4,-2,5,-2).

Cf. A002662 (1 X n), 2*A002662 (2 X n), A355552 (4 X n), A355553 (n X n).

LinearRecurrence[{4, -4, -2, 5, -2}, {1, 2, 8, 23, 61}, 50] (* Paolo Xausa, Oct 19 2024 *)

Python
```
def a(n): return 3*((1<
				
```

a(n) = 3*(2^n - 1 - n*(n+1)/2) + ceiling(n^2/2).
a(n) = A000982(n) + 3*A002662(n).
a(n) ~ 3*2^n.
From Stefano Spezia, Jul 10 2022: (Start)
G.f.: x*(1 - 2*x + 4*x^2 + x^3)/((1 - x)^3*(1 - x - 2*x^2)).
a(n) = (3*2^(n+2) - 4*n^2 - 6*n - 11 - (-1)^n)/4. (End)

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A355553 Number of ways to select 3 or more collinear points from an n X n grid.

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A355551 Number of ways to select 3 or more collinear points from a 3 X n grid.

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