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%I #37 Jun 14 2024 08:01:10
%S 1,7,13,25,43,61,85,115,145,181,223,265,313,367,421,481,547,613,685,
%T 763,841,925,1015,1105,1201,1303,1405,1513,1627,1741,1861,1987,2113,
%U 2245,2383,2521,2665,2815,2965,3121,3283,3445,3613,3787,3961,4141,4327,4513,4705,4903,5101,5305,5515,5725
%N Number of hexagonal lattice points within a hexagram centered at a lattice point and with outermost vertices at the six lattice points n steps outward from the central point.
%C In contrast, A003154 (the star numbers) counts the hexagonal lattice points within a hexagram centered at a lattice point and with the vertices of the central hexagon at the six lattice points a given number of steps outward from the central point.
%C Besides the first term, the first differences are given by six times A004396.
%C A005448 (the centered triangular numbers) counts just the lattice points within one of the two triangles that make up the hexagram.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F a(n) = 6*ceiling(n*(n+1)/3) + 1.
%F a(n) = 6*A007980(n-1) + 1 for n >= 1.
%F a(n+1) - a(n) = 6*A004396(n+1).
%F a(3n) = A081272(n).
%e Illustration of initial terms:
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%t Table[6*Ceiling[n*(n + 1)/3] + 1, {n, 0, 60}] (* _Amiram Eldar_, Jul 28 2023 *)
%o (PARI) a(n) = 6*ceil(n*(n+1)/3) + 1; \\ _Michel Marcus_, Jun 14 2024
%Y Cf. A003154, A004396, A005448, A007980, A081272.
%K nonn,easy
%O 0,2
%A _Aaron David Fairbanks_, May 17 2023