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

%I #32 Jan 17 2023 04:35:14

%S 5,10,23,54,117,240,497,1006,2027,4074,8169,16356,32741,65506,131039,

%T 262110,524253,1048536,2097113,4194262,8388563,16777170,33554385,

%U 67108812,134217677,268435402,536870855,1073741766,2147483589,4294967232,8589934529,17179869118,34359738299

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

%H Lucas A. Brown, <a href="/A355552/b355552.txt">Table of n, a(n) for n = 1..1000</a>

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A355552.py">A355552.py</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-1,-2,-1,2).

%F a(n) == H(n) + 3 * D4(n) + 2 * E(n), where

%F H(n) == 2^(n+2) - 4 - 2*n*(n+1),

%F D4(n) == floor((n^2 + 2) / 3), and

%F E(n) == floor((n^2 + 1) / 2).

%F a(n) ~ 2^(n+2).

%F 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

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

%K nonn,easy

%O 1,1

%A _Thomas Garrison_, Jul 14 2022

%E Corrected and extended by _Lucas A. Brown_, Oct 20 2022