A389579 Decreasing partition array based on the fractional parts of tan(n); see Comments.

1, 2, 3, 5, 6, 4, 8, 9, 18, 7, 11, 12, 21, 10, 16, 90, 15, 34, 13, 20, 17, 630, 93, 37, 14, 25, 26, 23, 1120, 445, 112, 19, 33, 40, 35, 29, 17092, 985, 140, 22, 53, 47, 38, 32, 36, 18292, 1874, 194, 24, 65, 49, 41, 39, 45, 51, 19065, 17447, 448, 27, 74, 50

Offset

1,2

Comments

Suppose that S = (s(m)), for m >= 1, is a sequence of distinct real numbers that is dense in an open interval (a,b), such as the numbers (sin(n)), for n>=1, dense in (-1,1). The increasing partition array (p(n,k)) of the set N of positive integers is defined inductively as follows: p(1,1) = 1, and for k >= 2, p(1,k) = least m such that s(m) > s(p(1,k-1)). For n>=2, p(n,1) = least new m (that is, m is not p(h,k) for any h<=n-1 and k>=1), and for k>=2, p(n,k) = least new m such that s(m) > s(p(n,k-1)).

The decreasing partition array (p(n,k)) of N is defined as follows: p(1,1)=s(1), and for k>=2, p(1,k) = least new m such that s(m) < s(p(1,k-1)). For n>=2, p(n,1) = least new m, and for k>=2, p(n,k) = least new m such that s(m) < (p(n,k-1)).

For a guide to related partition arrays, see A388853.

Links

Table of n, a(n) for n=1..61.

Example

Fractional parts of tan(n): 0.557..., -0.185..., -0.142..., 0.157..., -0.380..., -0.291..., etc., from which the first few terms of the increasing and decreasing partition arrays can be checked.
Corner:
   1   2   5   8  11  90  630 1120
   3   6   9  12  15  93  445  985
   4  18  21  34  37  112 140  194
   7  10  13  14  19  22   24   27
  16  20  25  33  53  65   74   81
  17  26  40  47  49  50   75   78
  23  35  38  41  44  46   55   72
  29  32  39  42  48  52   62   69
  36  45  57  58  60  63   66   68
  51  54  64  70  80  99  113  118

Mathematica

highs := {Map[First, #], Most[FoldList[Plus, 1, Map[Length, #]]]} &[
    Split[Rest[FoldList[Max, -\[Infinity], #]]]] &;
lows := {Map[First, #], Most[FoldList[Plus, 1, Map[Length, #]]]} &[
    Split[Rest[FoldList[Min, +\[Infinity], #]]]] &;
seqS = Table[N[FractionalPart[Tan[n]], 20], {n, 1, 1000}];
(* User:put your dense sequence S after seqS *)
indices = Range[Length[seqS]];
arrI = {};   (*start accumulating increasing partition array*)
Until[Last[arrI] == {}, AppendTo[arrI, Flatten[Map[Position[seqS, #] &,
     highs[seqS[[Complement[indices, Flatten[arrI]]]]][[1]]]]]];
Grid[Take[arrI, 12]]
arrD = {};   (*start accumulating decreasing partition array*)
Until[Last[arrD] == {}, AppendTo[arrD, Flatten[Map[Position[seqS, #] &,
     lows[seqS[[Complement[indices, Flatten[arrD]]]]][[1]]]]]];
Grid[Take[arrD, 12]]
(*_Peter J.C.Moses_, Sep 04 2025*)

Crossrefs

Cf. A388853, A389578.

Sequence in context: A137760 A368428 A360600 * A194863 A194833 A195108

Adjacent sequences: A389576 A389577 A389578 * A389580 A389581 A389582

Keyword

nonn,tabl

Author

Clark Kimberling, Oct 14 2025

Status

approved