

A191435


Dispersion of ([nx+n+x]), where x=(golden ratio) and [ ]=floor, by antidiagonals.


1



1, 5, 2, 15, 7, 3, 41, 20, 10, 4, 109, 54, 28, 13, 6, 287, 143, 75, 36, 18, 8, 753, 376, 198, 96, 49, 23, 9, 1973, 986, 520, 253, 130, 62, 26, 11, 5167, 2583, 1363, 664, 342, 164, 70, 31, 12, 13529, 6764, 3570, 1740, 897, 431, 185, 83, 34, 14, 35421, 17710
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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 nth 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.....5....15...41...109
2.....7....20...54...143
3.....10...28...75...198
4.....13...36...96...253
6.....18...49...130..342


MATHEMATICA

(* Program generates the dispersion array T of increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12; x = 1 + GoldenRatio;
f[n_] := Floor[n*x + x] (* f(n), 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}]] (* A191435 sequence *)


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KEYWORD



AUTHOR



STATUS

approved



