A248514 Fractal sequence of the dispersion of the "odious numbers".

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

Offset: 1

Author

Clark Kimberling, Oct 08 2014

Keywords

• nonn
• easy

Comments

As a fractal sequence, it contains infinitely many copies of itself: removing the first occurrence of each number leaves the original sequence.

Examples

			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, ...).
		

References

• Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168.

Links

• Clark Kimberling, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A248513.

Programs

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 *)