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A339483 Number of regular polygons that can be drawn with vertices on a centered hexagonal grid with side length n.

%I A339483 #35 Aug 21 2025 09:56:02
%S A339483 0,9,75,294,810,1815,3549,6300,10404,16245,24255,34914,48750,66339,
%T A339483 88305,115320,148104,187425,234099,288990,353010,427119,512325,609684,
%U A339483 720300,845325,985959,1143450,1319094,1514235,1730265,1968624,2230800,2518329,2832795
%N A339483 Number of regular polygons that can be drawn with vertices on a centered hexagonal grid with side length n.
%C A339483 The only regular polygons that can be drawn with vertices on the centered hexagonal grid are equilateral triangles and regular hexagons.
%H A339483 Peter Kagey, <a href="/A339483/b339483.txt">Table of n, a(n) for n = 0..10000</a>
%H A339483 Burkard Polster, <a href="https://youtu.be/sDfzCIWpS7Q">What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented</a>, Mathologer video (2020).
%H A339483 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A339483 a(n) = A000537(n) + A008893(n).
%F A339483 a(n) = (1/2)*(n+1)*n*(2*n+1)^2.
%F A339483 a(n) = 3*A180324(n).
%F A339483 Sum_{n>=1} 1/a(n) = 10 - Pi^2 (A348670). - _Amiram Eldar_, Jun 20 2025
%F A339483 From _Elmo R. Oliveira_, Aug 20 2025: (Start)
%F A339483 G.f.: -3*x*(x + 3)*(3*x + 1)/(x - 1)^5.
%F A339483 E.g.f.: exp(x)*x*(2 + x)*(9 + 24*x + 4*x^2)/2.
%F A339483 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F A339483 a(n) = A185096(n)/4 = A322677(n)/32. (End)
%e A339483 There are a(2) = 75 regular polygons that can be drawn on the centered hexagonal grid with side length 2: A000537(2) = 9 regular hexagons and A008893(n) = 66 equilateral triangles.
%e A339483 The nine hexagons are:
%e A339483     * . *       . * .       * * .
%e A339483    . . . .     * . . *     * . * .
%e A339483   * . . . *   . . . . .   . * * . .
%e A339483    . . . .     * . . *     . . . .
%e A339483     * . *       . * .       . . .
%e A339483       1           1           7
%e A339483 which are marked with the number of ways to draw the hexagons up to translation.
%e A339483 The 66 equilateral triangles are:
%e A339483     * . .       * . .       * . .       * . *       * . .       . . .
%e A339483    * * . .     . . * .     . . . .     . . . .     . . . .     * . . *
%e A339483   . . . . .   . * . . .   . . . * .   . . * . .   . . . . *   . . . . .
%e A339483    . . . .     . . . .     * . . .     . . . .     . . . .     . . . .
%e A339483     . . .       . . .       . . .       . . .       * . .       . * .
%e A339483      24          14          12          12           2           2
%e A339483 which are marked with the number of ways to draw the triangles up to translation and dihedral action of the hexagon.
%t A339483 a[n_] := n*(n+1)*(2*n+1)^2/2; Array[a, 35, 0] (* _Amiram Eldar_, Jun 20 2025 *)
%Y A339483 Cf. A000537 (regular hexagons), A008893 (equilateral triangles).
%Y A339483 Cf. A338323 (cubic grid).
%Y A339483 Cf. A003215, A180324, A185096, A322677, A348670.
%K A339483 nonn,easy
%O A339483 0,2
%A A339483 _Peter Kagey_, Dec 06 2020