A355553 -id:A355553 - cp's OEIS frontend

A355552 Number of ways to select 3 or more collinear points from a 4 X n grid.

5, 10, 23, 54, 117, 240, 497, 1006, 2027, 4074, 8169, 16356, 32741, 65506, 131039, 262110, 524253, 1048536, 2097113, 4194262, 8388563, 16777170, 33554385, 67108812, 134217677, 268435402, 536870855, 1073741766, 2147483589, 4294967232, 8589934529, 17179869118, 34359738299

Offset: 1

Thomas Garrison, Jul 14 2022

Lucas A. Brown, Table of n, a(n) for n = 1..1000
Lucas A. Brown, A355552.py
Index entries for linear recurrences with constant coefficients, signature (2,1,-1,-2,-1,2).

Cf. A000982 (1 X n), 2*A000982 (2 X n), A355553 (n X n).

a(n) == H(n) + 3 * D4(n) + 2 * E(n), where
H(n) == 2^(n+2) - 4 - 2*n*(n+1),
D4(n) == floor((n^2 + 2) / 3), and
E(n) == floor((n^2 + 1) / 2).
a(n) ~ 2^(n+2).
G.f.: -x * (6x^4 + 3x^3 - 2x^2 + 5) / ( (x - 1)^2 * (2x^2 + x - 1) * (x^2 + x + 1) ). - Lucas A. Brown, Oct 22 2022

Corrected and extended by Lucas A. Brown, Oct 20 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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A355552 Number of ways to select 3 or more collinear points from a 4 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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