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The 60-degree spoke (or ray) of a hexagonal spiral of Ulam.
7

%I #23 Oct 10 2022 07:56:28

%S 1,10,43,100,181,286,415,568,745,946,1171,1420,1693,1990,2311,2656,

%T 3025,3418,3835,4276,4741,5230,5743,6280,6841,7426,8035,8668,9325,

%U 10006,10711,11440,12193,12970,13771,14596,15445,16318,17215,18136,19081,20050,21043,22060,23101,24166,25255

%N The 60-degree spoke (or ray) of a hexagonal spiral of Ulam.

%H Colin Barker, <a href="/A244802/b244802.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F See A056105 example section for a formula.

%F From _Colin Barker_, Dec 12 2016: (Start)

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.

%F G.f.: x*(1 + 7*x + 16*x^2) / (1 - x)^3.

%F (End)

%e See A056105 example section for a diagram.

%p A244802:=n->12*n^2-27*n+16: seq(A244802(n), n=1..50); # _Wesley Ivan Hurt_, Jul 06 2014

%t f[n_] := 12n^2 - 27n + 16; Array[f, 47]

%o (PARI) vector(50, n, 12*n^2 - 27*n + 16) \\ _Michel Marcus_, Jul 06 2014

%o (PARI) Vec(x*(1 + 7*x + 16*x^2) / (1 - x)^3 + O(x^50)) \\ _Colin Barker_, Dec 12 2016

%o (Magma) [12*n^2-27*n+16 : n in [1..50]]; // _Wesley Ivan Hurt_, Jul 06 2014

%Y Cf. A056105, A056106, A244803, A056107, A244804, A056108, A244805, A056109, A244806, A003215.

%K nonn,easy

%O 1,2

%A _Robert G. Wilson v_, Jul 06 2014