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 A243712 Irregular triangular array of denominators of all positive rational numbers ordered as in Comments. 12
 1, 1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 3, 1, 2, 3, 4, 5, 3, 6, 5, 5, 1, 2, 3, 4, 5, 3, 6, 5, 5, 7, 7, 8, 7, 1, 2, 3, 4, 5, 3, 6, 5, 5, 7, 7, 8, 7, 8, 9, 11, 11, 9, 4, 1, 2, 3, 4, 5, 3, 6, 5, 5, 7, 7, 8, 7, 8, 9, 11, 11, 9, 4, 9, 11, 14, 15, 14, 7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Decree that (row 1) = (1), (row 2) = (2), and (row 3) = (3). Thereafter, row n consists of the following numbers arranged in decreasing order: 1 + x for each x in (row n-1), together with x/(x + 1) for each x in row (n-3). Every positive rational number occurs exactly once in the array. The number of numbers in (row n) is A000930(n-1), for n >= 1. LINKS Clark Kimberling, Table of n, a(n) for n = 1..1000 EXAMPLE First 8 rows of the array of all positive rationals: 1/1 2/1 3/1 4/1 .. 1/2 5/1 .. 3/2 .. 2/3 6/1 .. 5/2 .. 5/3 ... 3/4 7/1 .. 7/2 .. 8/3 ... 7/4 ... 4/5 .. 1/3 8/1 .. 9/2 .. 11/3 .. 11/4 .. 9/5 .. 4/3 .. 5/6 .. 3/5 .. 2/5 The denominators, by rows: 1,1,1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,3,1,2,3,4,5,3,6,5,5,... MATHEMATICA z = 13; g[1] = {1}; f1[x_] := x + 1; f2[x_] := -1/x; h[1] = g[1]; b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]]; h[n_] := h[n] = Union[h[n - 1], g[n - 1]]; g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]; u = Table[g[n], {n, 1, z}]; u1 = Delete[Flatten[u], 10] w[1] = 0; w[2] = 1; w[3] = 1; w[n_] := w[n - 1] + w[n - 3]; u2 = Table[Drop[g[n], w[n]], {n, 1, z}]; u3 = Delete[Delete[Flatten[Map[Reverse, u2]], 4], 4] Denominator[u3] (* A243712 *) Numerator[u3] (* A243713 *) Denominator[u1] (* A243714 *) Numerator[u1] (* A243715 *) CROSSREFS Cf. A243713, A243714, A243715, A000930, A243613. Sequence in context: A133994 A066041 A194965 * A256553 A194896 A212721 Adjacent sequences: A243709 A243710 A243711 * A243713 A243714 A243715 KEYWORD nonn,easy,tabf,frac AUTHOR Clark Kimberling, Jun 09 2014 STATUS approved

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Last modified April 22 02:54 EDT 2024. Contains 371887 sequences. (Running on oeis4.)