

A191426


Dispersion of (3+[nr]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.


91



1, 4, 2, 9, 6, 3, 17, 12, 7, 5, 30, 22, 14, 11, 8, 51, 38, 25, 20, 15, 10, 85, 64, 43, 35, 27, 19, 13, 140, 106, 72, 59, 46, 33, 24, 16, 229, 174, 119, 98, 77, 56, 41, 28, 18, 373, 284, 195, 161, 127, 93, 69, 48, 32, 21, 606, 462, 318, 263, 208, 153, 114, 80, 54, 36, 23, 983, 750, 517, 428, 339, 250, 187, 132, 90, 61, 40, 26
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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:
(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: A191426A191455.


REFERENCES

Clark Kimberling, Fractal sequences and interspersions, Ars Combinatoria 45 (1997) 157168.


LINKS

Table of n, a(n) for n=1..78.
Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3, Fall 1998, p. 176. Solution published in Vol. 12, No. 1, Winter 2000, pp. 6162.
Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313321.


EXAMPLE

Northwest corner:
1...4...9...17..30
2...6...12..22..38
3...7...14..25..43
5...11..20..35..59
8...15..27..46..77


MATHEMATICA

(* Program generates the dispersion array T of increasing sequence f[n] *)
r = 40; r1 = 12; (* r=#rows of T, r1=#rows to show *)
c = 40; c1 = 12; (* c=#cols of T, c1=#cols to show *)
x = GoldenRatio; f[n_] := Floor[n*x + 3]
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}]]
(* A191426 array *)
Flatten[Table[t[k, n  k + 1], {n, 1, c1}, {k, 1, n}]] (* A191426 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)


CROSSREFS

Cf. A114537, A035513, A035506.
Sequence in context: A285090 A114577 A285089 * A182702 A282600 A326845
Adjacent sequences: A191423 A191424 A191425 * A191427 A191428 A191429


KEYWORD

nonn,tabl


AUTHOR

Clark Kimberling, Jun 02 2011


STATUS

approved



