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Number of distinct sums of n complex 6th power roots of unity.
1

%I #16 Nov 03 2024 17:46:01

%S 1,6,19,37,61,91,127,169,217,271,331,397,469,547,631,721,817,919,1027,

%T 1141,1261,1387,1519,1657,1801,1951,2107,2269,2437,2611,2791,2977,

%U 3169,3367,3571,3781,3997,4219,4447,4681,4921,5167,5419,5677,5941,6211,6487,6769,7057,7351,7651,7957

%N Number of distinct sums of n complex 6th power roots of unity.

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

%F For n >= 2, a(n) = 3*n^2 + 3*n + 1 = A003215(n).

%F For n >= 5, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%F G.f. (1 + 3*x + 4*x^2 - 3*x^3 + x^4) / (1 - x)^3.

%t LinearRecurrence[{3,-3,1},{1,6,19,37,61},60] (* _Harvey P. Dale_, Nov 03 2024 *)

%o (PARI) a(n)=if(n==1, 6, 3*n*(n+1)+1) \\ _Charles R Greathouse IV_, Aug 15 2022

%Y Same as A003215 except for a(1) = 6.

%Y Row 6 of A299807.

%Y Cf. A000012, A000027, A000217, A000290, A000332, A000579, A014820, A103314, A107754, A107861, A108380, A107848, A107753, A108381, A143008, A299754.

%K nonn,easy

%O 0,2

%A _Max Alekseyev_, Aug 15 2022