A090024 - OEIS

%I #18 Mar 12 2022 07:57:42

%S 0,1,45,571,5841,55651,515025,4702531,42649281,385447171,3476958705,

%T 31332052291,282184860321,2540643522691,22870684139985,

%U 205860600134851,1852867557848961,16676418630942211,150090820212050865

%N Number of distinct lines through the origin in the n-dimensional lattice of side length 8.

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

%H Gennady Eremin, <a href="/A090024/b090024.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (20,-140,430,-579,270).

%F a(n) = 9^n - 5^n - 3^n - 2^n + 2.

%F G.f.: -x*(291*x^3-189*x^2+25*x+1)/((x-1)*(2*x-1)*(3*x-1)*(5*x-1)*(9*x-1)). [_Colin Barker_, Sep 04 2012]

%e a(2) = 45 because in 2D the lines have slope 0, 1/8, 3/8, 5/8, 7/8, 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 1/6, 5/6, 1/5, 2/5, 3/5, 4/5, 1/4, 3/4, 1/3, 2/3, 1/2, 1 and their reciprocals.

%t Table[9^n - 5^n - 3^n - 2^n + 2, {n, 0, 20}]

%o (Python)

%o [9**n-5**n-3**n-2**n+2 for n in range(30)] # _Gennady Eremin_, Mar 12 2022

%Y a(n) = T(n, 5) from A090030. Cf. A000225, A001047, A060867, A090020, A090021, A090022, A090023 are for dimension n with side lengths 1, 2, 3, 4, 5, 6, 7 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 20 2003