A389585 Decreasing partition array based on the fractional parts of (n*sqrt(5)); see A388853.

1, 5, 2, 9, 6, 3, 13, 10, 7, 4, 17, 14, 11, 8, 21, 89, 18, 15, 12, 25, 38, 161, 22, 19, 16, 29, 42, 55, 233, 26, 23, 20, 33, 46, 59, 72, 305, 30, 27, 24, 37, 50, 63, 76, 93, 1597, 34, 31, 28, 41, 54, 67, 80, 97, 110, 2889, 106, 35, 32, 45, 58, 71, 84, 101

Offset

1,2

Links

Table of n, a(n) for n=1..64.

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    5    9   13   17   89  161  233  305
     2    6   10   14   18   22   26   30   34
     3    7   11   15   19   23   27   31   35
     4    8   12   16   20   24   28   32   36
    21   25   29   33   37   41   45   49   53
    38   42   46   50   54   58   62   66   70
    55   59   63   67   71   75   79   83   87
    72   76   80   84   88   92   96  100  104
    93   97  101  105  109  113  117  121  125

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, A389584.

Sequence in context: A358516 A177345 A209304 * A367844 A258503 A180726

Adjacent sequences: A389582 A389583 A389584 * A389586 A389587 A389588

Keyword

nonn,tabl

Author

Clark Kimberling, Oct 19 2025

Status

approved