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A191538
Dispersion of (4*n-floor(n*sqrt(3))), by antidiagonals.
1
1, 3, 2, 7, 5, 4, 16, 12, 10, 6, 37, 28, 23, 14, 8, 84, 64, 53, 32, 19, 9, 191, 146, 121, 73, 44, 21, 11, 434, 332, 275, 166, 100, 48, 25, 13, 985, 753, 624, 377, 227, 109, 57, 30, 15, 2234, 1708, 1416, 856, 515, 248, 130, 69, 35, 17, 5067, 3874, 3212, 1942
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 and A191536-A191545.
EXAMPLE
Northwest corner:
1, 3, 7, 16, 37, ...
2, 5, 12, 28, 64, ...
4, 10, 23, 53, 121, ...
6, 14, 32, 73, 166, ...
8, 19, 44, 100, 227, ...
MATHEMATICA
(* Program generates the dispersion array T of the increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] :=4n-Floor[n*Sqrt[3]] (* 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, r1}, {j, 1, c1}]] (* A191538 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191538 sequence *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jun 06 2011
STATUS
approved