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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 (list; table; graph; refs; listen; history; text; internal format)
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.

REFERENCES

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

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. 61-62.

Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.

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 A257734

Adjacent sequences:  A191423 A191424 A191425 * A191427 A191428 A191429

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jun 02 2011

STATUS

approved

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Last modified June 28 03:21 EDT 2017. Contains 288813 sequences.