|
| |
|
|
A248514
|
|
Fractal sequence of the dispersion of the "odious numbers".
|
|
2
|
|
|
|
1, 1, 1, 2, 1, 3, 4, 2, 1, 5, 6, 3, 7, 4, 2, 8, 1, 9, 10, 5, 11, 6, 3, 12, 13, 7, 4, 14, 2, 15, 16, 8, 1, 17, 18, 9, 19, 10, 5, 20, 21, 11, 6, 22, 3, 23, 24, 12, 25, 13, 7, 26, 4, 27, 28, 14, 2, 29, 30, 15, 31, 16, 8, 32, 1, 33, 34, 17, 35, 18, 9, 36, 37, 19
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,4
|
|
|
COMMENTS
|
As a fractal sequence, it contains infinitely many copies of itself: removing the first occurrence of each number leaves the original sequence.
|
|
|
REFERENCES
|
Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
A northwest corner of the dispersion (A248513) of the "odious numbers" (A181155) follows:
1 ... 2 ... 3 ... 5 ... 9 ... 17 .... 33
4 ... 8 ... 15 .. 29 .. 57 .. 113 ... 225
6 ... 12 .. 23 .. 45 .. 89 .. 177 ... 353
7 ... 14 .. 27 .. 53 .. 105 .. 209 .. 417
10 .. 20 .. 39 .. 77 .. 153 .. 305 .. 609
The numbers 1, 2, 3, 4, 5 appear in rows 1, 1, 1, 2, 1, respectively, so that A248514 = (1, 1, 1, 2, 1, ...).
|
|
|
MATHEMATICA
|
r = 40; r1 = 10; (* r = # rows of T, r1 = # rows to show*);
c = 40; c1 = 12; (* c = # cols of T, c1 = # cols to show*);
x = GoldenRatio; s[n_] := s[n] = If[n < 1, 0, 2 n - Mod[Total[IntegerDigits[n - 1, 2]], 2]];
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]; rows = {NestList[s, 1, c]};
Do[rows = Append[rows, NestList[s, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]]; TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]] (* A248513 array*)
u = Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A248013 sequence*)
row[i_] := row[i] = Table[t[i, j], {j, 1, c}]
f[n_] := Select[Range[r], MemberQ[row[#], n] &]
v = Flatten[Table[f[n], {n, 1, 200}]] (* A248514 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
