|
|
A191446
|
|
Dispersion of [n*sqrt(5)], where [ ]=floor, by antidiagonals.
|
|
2
|
|
|
1, 2, 3, 4, 6, 5, 8, 13, 11, 7, 17, 29, 24, 15, 9, 38, 64, 53, 33, 20, 10, 84, 143, 118, 73, 44, 22, 12, 187, 319, 263, 163, 98, 49, 26, 14, 418, 713, 588, 364, 219, 109, 58, 31, 16, 934, 1594, 1314, 813, 489, 243, 129, 69, 35, 18, 2088, 3564, 2938, 1817
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Background discussion: Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1. The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n)), s(s(s(t(n)))), ...). Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers. The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence. Examples:
|
|
LINKS
|
|
|
EXAMPLE
|
Northwest corner:
1...2....4....8...17
3...6....13...29..64
5...11...24...53..118
7...15...33...73..163
9...20...44...98..219
|
|
MATHEMATICA
|
(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; x = Sqrt[5];
f[n_] := Floor[n*x] (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191446 sequence *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Corrected typo in name and fixed Mathematica program by Vaclav Kotesovec, Oct 24 2014
|
|
STATUS
|
approved
|
|
|
|