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A262147
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Numerators of domain values whose images via the flowsnake Q-function follow a spiral trajectory (see comments).
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1
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3, 13, 115, 125, 19, 141, 1011, 1021, 7171, 7181, 1027, 7197, 50403, 50413, 352915, 352925, 50419, 352941, 2470611, 2470621, 17294371, 17294381, 2470627, 17294397, 121060803, 121060813, 847425715, 847425725, 121060819, 847425741
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OFFSET
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1,1
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COMMENTS
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The spiral trajectory joins the centroids of hexagons that fit into a substitution hierarchy. Assign to each level of the hierarchy an index N in {0, 1, 2, ... }. Level 0 is a single hexagon. Subsequent levels of the hierarchy are enumerated by subdividing each hexagon at level N into seven hexagons at level N+1 (Cf. Gardner ). Call a hexagon central if its centroid coincides with the centroid of the hexagon at level 0. We determine a set of points in the plane by taking, for each N, the six centroids at level N+1 that fall within the central hexagon at level N but do not fall within the central hexagon at level N+1. Centroids are rational single-points, so the Q-function determines a bijection between our set of plane points and a set of rational numbers ( Klee, Complex Systems, Wolfram Demonstrations ). The fractional sequence a(n) is that set of rational numbers ordered by a(n+1) > a(n). If we plot the image of a(n) we only see the spiral over domain [0,3/8] as it converges to the center of Gosper Island. The complete, double-spiral appears as Figure 12 in "A Pit of Flowsnakes".
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REFERENCES
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M. Gardner, Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, Cambridge University Press, 1997, pages 36-42.
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LINKS
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FORMULA
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a(n) = 50*a(n-6)-49*a(n-12) for n>12.
G.f.: (3 +13*x +115*x^2 +125*x^3 +19*x^4 +141*x^5 +861*x^6 +371*x^7 +1421*x^8 +931*x^9 +77*x^10 +147*x^11) / ((1 -x)*(1 +x)*(1 -x +x^2)*(1 +x +x^2)*(1 -7*x^3)*(1 +7*x^3)).
(End)
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EXAMPLE
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FLSN[{3/56, 13/56, 115/392, 125/392, 19/56, 141/392, 1011/2744, 1021/2744, 7171/19208, 7181/19208, 1027/2744, 7197/19208}] = {3/14 + x/42, 4/7 - x/21, 30/49 + (17 x)/147, 31/49 + (29 x)/147, 51/98 + (73 x)/294, 19/49 + (32 x)/147, 153/343 + (170 x)/1029, 163/343 + (143 x)/1029, 1212/2401 + (1118 x)/7203, 2495/4802 + (2353 x)/14406, 1236/2401 + (1259 x)/7203, 1189/2401 + (1283 x)/7203}, where x = ( sqrt(3) * i ).
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MATHEMATICA
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b[n_] := Select[Union@Join[Flatten[MapIndexed[#1/(8 7^#2[[1]]) &, MapIndexed[With[{l =If[OddQ[#2[[1]] ], {10, 14, 8, 8, 10}, {10, 8, 8, 14, 10}]}, FoldList[Plus, #1, l]] &, FoldList[7 (#1 + #2) + 24 &, 3, Table[If[OddQ[i], 10, 26], {i, 1, 2 Ceiling[n/6]}]] ] ] ] ], # < 3/8 &][[1 ;; n]]; Numerator[b[100]]
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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