A090020 - OEIS

%I #18 Mar 04 2022 17:08:12

%S 0,1,13,91,529,2851,14833,75811,383809,1932931,9705553,48648931,

%T 243605089,1219100611,6098716273,30503196451,152544778369,

%U 762810181891,3814309582993,19072323542371,95363943807649,476826695752771

%N Number of distinct lines through the origin in the n-dimensional lattice of side length 4.

%C Equivalently, lattice points where the gcd of all the coordinates is 1.

%H Indranil Ghosh, <a href="/A090020/b090020.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (11,-41,61,-30).

%F a(n) = 5^n - 3^n - 2^n + 1.

%F G.f.: -x*(11*x^2-2*x-1)/((x-1)*(2*x-1)*(3*x-1)*(5*x-1)). [_Colin Barker_, Sep 04 2012]

%e a(2) = 13 because in 2D the lines have slope 0, 1/4, 1/3, 1/2, 2/3, 3/4, 1, 4/3, 3/2, 2, 3, 4 and infinity.

%t Table[5^n - 3^n - 2^n + 1, {n, 0, 25}]

%t LinearRecurrence[{11,-41,61,-30},{0,1,13,91},30] (* _Indranil Ghosh_, Feb 21 2017 *)

%o (Python) def A090020(n): return 5**n-3**n-2**n+1 # _Indranil Ghosh_, Feb 21 2017

%Y a(n) = T(n,4) from A090030. Cf. A000225, A001047, A060867, A090021, A090022, A090023, A090024 are for dimension n with side lengths 1, 2, 3, 5, 6, 7, 8 respectively. A049691, A090025, A090026, A090027, A090028, A090029 are for side length k in 2, 3, 4, 5, 6, 7 dimensions.

%K easy,nonn

%O 0,3

%A _Joshua Zucker_, Nov 19 2003