A389584 Increasing partition array based on the fractional parts of (n*sqrt(5)); see A388853.

1, 2, 5, 3, 6, 9, 4, 7, 10, 13, 21, 8, 11, 14, 17, 38, 25, 12, 15, 18, 22, 55, 42, 29, 16, 19, 23, 26, 72, 59, 46, 33, 20, 24, 27, 30, 377, 76, 63, 50, 37, 41, 28, 31, 34, 682, 93, 80, 67, 54, 58, 45, 32, 35, 39, 987, 110, 97, 84, 71, 75, 62, 49, 36, 40, 43

Offset

1,2

Links

Table of n, a(n) for n=1..66.

Example

Fractional parts of n*sqrt(5): 0.236..., 0.472..., 0.708..., 0.944..., 0.180..., 0.416..., etc., from which the first few terms of the increasing and decreasing partition arrays can be checked.
Corner:
    1    2    3    4   21   38   55   72  377
    5    6    7    8   25   42   59   76   93
    9   10   11   12   29   46   63   80   97
   13   14   15   16   33   50   67   84  101
   17   18   19   20   37   54   71   88  105
   22   23   24   41   58   75   92  109  126
   26   27   28   45   62   79   96  113  130
   30   31   32   49   66   83  100  117  134

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[n*Sqrt[5]], 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, A389585.

Sequence in context: A256997 A335499 A358838 * A359008 A239970 A111202

Adjacent sequences: A389581 A389582 A389583 * A389585 A389586 A389587

Keyword

nonn,tabl

Author

Clark Kimberling, Oct 19 2025

Status

approved