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A090021 Number of distinct lines through the origin in the n-dimensional lattice of side length 5. 11

%I

%S 0,1,21,175,1185,7471,45801,277495,1672545,10056991,60405081,

%T 362615815,2176242705,13059083311,78359348361,470170570135,

%U 2821066729665,16926530042431,101559568723641,609358576700455,3656154951181425

%N Number of distinct lines through the origin in the n-dimensional lattice of side length 5.

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

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (12,-47,72,-36).

%F a(n) = 6^n - 3^n - 2*2^n + 2.

%F G.f.: -x*(30*x^2-9*x-1)/((x-1)*(2*x-1)*(3*x-1)*(6*x-1)). [_Colin Barker_, Sep 04 2012]

%e a(2) = 21 because in 2D the lines have slope 0, 1/5, 2/5, 3/5, 4/5, 1/4, 3/4, 1/3, 2/3, 1/2, 1 and their reciprocals.

%t Table[6^n - 3^n - 2*2^n + 2, {n, 0, 25}]

%t LinearRecurrence[{12,-47,72,-36},{0,1,21,175},30] (* _Harvey P. Dale_, Jul 18 2016 *)

%Y a(n) = T(n, 5) from A090030. Cf. A000225, A001047, A060867, A090020, A090022, A090023, A090024 are for dimension n with side lengths 1, 2, 3, 4, 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

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Last modified July 29 09:41 EDT 2021. Contains 346344 sequences. (Running on oeis4.)