OFFSET
1,2
EXAMPLE
Fractional parts of e*n: 0.718..., 0.436..., 0.154..., 0.873..., 0.591..., 0.309..., 0.027..., etc., from which the first few terms of the increasing and decreasing partition arrays can be checked.
Corner:
1 2 3 7 39 110 181 252 323 394 465
4 5 6 10 14 46 78 149 220 291 362
8 9 13 17 21 53 85 117 188 259 330
11 12 16 20 24 28 60 92 124 156 227
15 19 23 27 31 35 67 99 131 163 195
18 22 26 30 34 38 42 74 106 138 170
25 29 33 37 41 45 49 81 113 145 177
32 36 40 44 48 52 56 88 120 152 184
43 47 51 55 59 63 95 127 159 191 223
50 54 58 62 66 70 102 134 166 198 230
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[E*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
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Oct 15 2025
STATUS
approved
