|
%I #20 Mar 09 2022 10:38:28
%S 0,1,37,415,3745,31471,257257,2078455,16704865,133935391,1072633177,
%T 8585561095,68702163985,549687102511,4397773276297,35183283965335,
%U 281470638631105,2251782504544831,18014329402322617,144114912035163175,1152920401607386225
%N Number of distinct lines through the origin in the n-dimensional lattice of side length 7.
%C Equivalently, lattice points where the gcd of all the coordinates is 1.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (18,-115,330,-424,192).
%F a(n) = 8^n - 4^n - 3^n - 2^n + 2.
%F G.f.: -x*(200*x^3-136*x^2+19*x+1)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(8*x-1)). - _Colin Barker_, Sep 04 2012
%e a(2) = 37 because in 2D the lines have slope 0, 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[8^n - 4^n - 3^n - 2^n + 2, {n, 0, 20}]
%o (Python)
%o [8**n-4**n-3**n-2**n+2 for n in range(25)] # _Gennady Eremin_, Mar 09 2022
%Y Equals A090030(n+7,n).
%Y Cf. A000225, A001047, A060867, A090020, A090021, A090022, A090024 are for dimension n with side lengths 1, 2, 3, 4, 5, 6, 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 20 2003
|
