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A191545
Dispersion of (floor(9*n/4)), by antidiagonals.
6
1, 2, 3, 4, 6, 5, 9, 13, 11, 7, 20, 29, 24, 15, 8, 45, 65, 54, 33, 18, 10, 101, 146, 121, 74, 40, 22, 12, 227, 328, 272, 166, 90, 49, 27, 14, 510, 738, 612, 373, 202, 110, 60, 31, 16, 1147, 1660, 1377, 839, 454, 247, 135, 69, 36, 17, 2580, 3735, 3098, 1887
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:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.
EXAMPLE
Northwest corner:
1...2....4....9....20
3...6....13...29...65
5...11...25...54...121
7...15...33...74...166
8...18...40...90...202
MATHEMATICA
(* Program generates the dispersion array T of the complement of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] :=Floor[9n/4]] (* 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}]]
(* A191545 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191545 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)
CROSSREFS
KEYWORD
nonn,tabl,changed
AUTHOR
Clark Kimberling, Jun 09 2011
STATUS
approved