A133299 -id:A133299 - cp's OEIS frontend

A191429 Dispersion of ([n*sqrt(2)+2]), where [ ]=floor, by antidiagonals.

1, 3, 2, 6, 4, 5, 10, 7, 9, 8, 16, 11, 14, 13, 12, 24, 17, 21, 20, 18, 15, 35, 26, 31, 30, 27, 23, 19, 51, 38, 45, 44, 40, 34, 28, 22, 74, 55, 65, 64, 58, 50, 41, 33, 25, 106, 79, 93, 92, 84, 72, 59, 48, 37, 29, 151, 113, 133, 132, 120, 103, 85, 69, 54, 43, 32, 215, 161, 190, 188, 171, 147, 122, 99, 78, 62, 47, 36

Offset: 1

Clark Kimberling, Jun 03 2011

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.

			Northwest corner:
  1...3...6...10..16
  2...4...7...11..17
  5...9...14..21..31
  8...13..20..30..44
  12..18..27..40..58

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r = 40; r1 = 12;  (* r=# rows of T to compute, r1=# rows to show *)
c = 40; c1 = 12;   (* c=# cols to compute, c1=# cols to show *)
x = Sqrt[2];
f[n_] := Floor[n*x + 2] (* f(n) is 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]];  (* the array T *)
TableForm[
Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191429 array *)
Flatten[Table[
  t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191429 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191432 Dispersion of ([n*x+1/x]), where x=sqrt(2) and [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 2, 5, 3, 7, 8, 4, 10, 12, 11, 6, 14, 17, 16, 15, 9, 20, 24, 23, 21, 18, 13, 28, 34, 33, 30, 26, 22, 19, 40, 48, 47, 43, 37, 31, 25, 27, 57, 68, 67, 61, 53, 44, 36, 29, 38, 81, 96, 95, 86, 75, 62, 51, 41, 32, 54, 115, 136, 135, 122, 106, 88, 72, 58, 45, 35, 77, 163, 193, 191, 173, 150, 125, 102, 82, 64, 50, 39

Offset: 1

Clark Kimberling, Jun 03 2011

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.
Conjecture: It appears this sequence is related to the even numbers with odd abundance A088827. Looking at the table format if the columns represent the powers of 2 (starting at 2^1) and the rows represent the squares of odd numbers, then taking the product of a term's row and column gives the n-th term in A088827. Example: A088827(67) = (7^2) * (2^6) = 3136. - John Tyler Rascoe, Jul 12 2022

			Northwest corner:
   1    2    3    4    6    9
   5    7   10   14   20   28
   8   12   17   24   34   48
  11   16   23   33   47   67
  15   21   30   43   61   86

Michel Marcus, Antidiagonals n = 1..100, flattened

Cf. A114537, A035513, A035506, A088827.

Cf. A001953, A001954.

(* 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 = Sqrt[2];
f[n_] := Floor[n*x + 1/x] (* f(n) is 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}]] (* A191432 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191432 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

s(n) = my(x=quadgen(8)); floor(n*x+1/x);  \\ A001953
t(n) = floor((n+1/2)*(2+quadgen(8))); \\ A001954
T(n, k) = my(x = t(n-1)); for (i=2, k, x = s(x);); x; \\ Michel Marcus, Jul 13 2022

A191436 Dispersion of ([n*x+n+x-1]), where x=(golden ratio) and [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 4, 2, 12, 6, 3, 33, 17, 9, 5, 88, 46, 25, 14, 7, 232, 122, 67, 38, 19, 8, 609, 321, 177, 101, 51, 22, 10, 1596, 842, 465, 266, 135, 59, 27, 11, 4180, 2206, 1219, 698, 355, 156, 72, 30, 13, 10945, 5777, 3193, 1829, 931, 410, 190, 80, 35, 15, 28656, 15126

Offset: 1

Clark Kimberling, Jun 04 2011

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.

			Northwest corner:
  1....4....12...33...88
  2....6....17...46...122
  3....9....25...67...177
  5....14...38...101..266
  7....19...51...135..355

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;  x = GoldenRatio;
f[n_] := Floor[n*x+n+x-1] (* 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}]]
(* A191436 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191436 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191439 Dispersion of ([n*sqrt(2)+n+1/2]), where [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 7, 4, 12, 17, 10, 6, 29, 41, 24, 14, 8, 70, 99, 58, 34, 19, 9, 169, 239, 140, 82, 46, 22, 11, 408, 577, 338, 198, 111, 53, 27, 13, 985, 1393, 816, 478, 268, 128, 65, 31, 15, 2378, 3363, 1970, 1154, 647, 309, 157, 75, 36, 16, 5741, 8119, 4756

Offset: 1

Clark Kimberling, Jun 04 2011

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.

			Northwest corner:
  1....2....5....12...29
  3....7....17...41...99
  4....10...24...58...140
  6....14...34...82...198
  8....19...46...111..268

Robbert Fokkink, The Pell Tower and Ostronometry, arXiv:2309.01644 [math.CO], 2023.

Cf. A114537, A035513, A035506, A191438.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;  x = Sqr[2];
f[n_] := Floor[n*x+n+1/2] (* 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}]]
(* A191439 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191439 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191446 Dispersion of [n*sqrt(5)], where [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 13, 11, 7, 17, 29, 24, 15, 9, 38, 64, 53, 33, 20, 10, 84, 143, 118, 73, 44, 22, 12, 187, 319, 263, 163, 98, 49, 26, 14, 418, 713, 588, 364, 219, 109, 58, 31, 16, 934, 1594, 1314, 813, 489, 243, 129, 69, 35, 18, 2088, 3564, 2938, 1817

Offset: 1

Clark Kimberling, Jun 05 2011

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.

			Northwest corner:
  1...2....4....8...17
  3...6....13...29..64
  5...11...24...53..118
  7...15...33...73..163
  9...20...44...98..219

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;  x = Sqrt[5];
f[n_] := Floor[n*x] (* 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}]]
(* A191446 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191446 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

Corrected typo in name and fixed Mathematica program by Vaclav Kotesovec, Oct 24 2014

A191540 Dispersion of (floor(2nsqrt(2))), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 8, 4, 14, 22, 11, 6, 39, 62, 31, 16, 7, 110, 175, 87, 45, 19, 9, 311, 494, 246, 127, 53, 25, 10, 879, 1397, 695, 359, 149, 70, 28, 12, 2486, 3951, 1965, 1015, 421, 197, 79, 33, 13, 7031, 11175, 5557, 2870, 1190, 557, 223, 93, 36, 15, 19886, 31607

Offset: 1

Clark Kimberling, Jun 06 2011

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.

			Northwest corner:
  1,  2,  5,  14,  39, ...
  3,  8, 22,  62, 175, ...
  4, 11, 31,  87, 246, ...
  6, 16, 45, 127, 359, ...
  7, 19, 53, 149, 421, ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] :=Floor[2n*Sqrt[2]]   (* 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}]]  (* A191540 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191540 sequence *)

A191427 Dispersion of ([n*r+3/2]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 11, 7, 9, 8, 19, 12, 16, 14, 10, 32, 20, 27, 24, 17, 13, 53, 33, 45, 40, 29, 22, 15, 87, 54, 74, 66, 48, 37, 25, 18, 142, 88, 121, 108, 79, 61, 41, 30, 21, 231, 143, 197, 176, 129, 100, 67, 50, 35, 23, 375, 232, 320, 286, 210, 163, 109, 82, 58, 38, 26, 608, 376, 519, 464, 341, 265, 177, 134, 95, 62, 43, 28

Offset: 1

Clark Kimberling, Jun 02 2011

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.

			Northwest corner:
  1...3...6...11..19
  2...4...7...12..20
  5...9...16..27..45
  8...14..24..40..66
  10..17..29..48..79

Cf. A114537, A035513, A035506.

(* 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 + 3/2]
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}]]
(* A191427 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
(* A191427 sequence *)
(* Peter J. C. Moses, Jun 01 2011 *)

A191428 Dispersion of ([n*r+r]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 11, 8, 9, 7, 19, 14, 16, 12, 10, 32, 24, 27, 21, 17, 13, 53, 40, 45, 35, 29, 22, 15, 87, 66, 74, 58, 48, 37, 25, 18, 142, 108, 121, 95, 79, 61, 42, 30, 20, 231, 176, 197, 155, 129, 100, 69, 50, 33, 23, 375, 286, 320, 252, 210, 163, 113, 82, 55, 38, 26, 608, 464, 519, 409, 341, 265, 184, 134, 90, 63, 43, 28

Offset: 1

Clark Kimberling, Jun 03 2011

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.

			Northwest corner:
  1...3...6...11..19
  2...4...8...14..24
  5...9...16..27..45
  7...12..21..35..58
  10..17..29..48..79

Cf. A114537, A035513, A035506.

(* 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 + x]
(* f(n) is 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}]]
(* A191428 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
(* A191428 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191430 Dispersion of ([n*sqrt(2)+3/2]), where [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 13, 17, 18, 21, 22, 19, 16, 25, 26, 31, 32, 28, 24, 20, 36, 38, 45, 46, 41, 35, 29, 23, 52, 55, 65, 66, 59, 50, 42, 34, 27, 75, 79, 93, 94, 84, 72, 60, 49, 39, 30, 107, 113, 133, 134, 120, 103, 86, 70, 56, 43, 33, 152, 161, 189, 191, 171, 147, 123, 100, 80, 62, 48, 37

Offset: 1

Clark Kimberling, Jun 03 2011

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.

			Northwest corner:
  1...2...4...7...11
  3...5...12..18..18
  6...9...14..21..31
  10..15..22..32..46
  13..19..28..41..59

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r = 40; r1 = 12;  (* r=# rows of T to compute, r1=# rows to show *)
c = 40; c1 = 12;   (* c=# cols to compute, c1=# cols to show *)
x = Sqrt[2];
f[n_] := Floor[n*x + 3/2] (* f(n) is 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]];  (* the array T *)
TableForm[
Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191430 array *)
Flatten[Table[
  t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191430 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191431 Dispersion of ([n*x+x]), where x=sqrt(2) and [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 13, 16, 18, 21, 22, 19, 17, 24, 26, 31, 32, 28, 25, 20, 35, 38, 45, 46, 41, 36, 29, 23, 50, 55, 65, 66, 59, 52, 42, 33, 27, 72, 79, 93, 94, 84, 74, 60, 48, 39, 30, 103, 113, 132, 134, 120, 106, 86, 69, 56, 43, 34, 147, 161, 188, 190, 171, 151, 123, 98, 80, 62, 49, 37

Offset: 1

Clark Kimberling, Jun 03 2011

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.

			Northwest corner:
  1.....2....4....7...11...16
  3.....5....8...12...18...26
  6.....9...14...21...31...45
  10...15...22...32...46...66
  13...19...28...41...59...84

Cf. A114537, A035513, A035506.

(* 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 = Sqrt[2];
f[n_] := Floor[n*x + x] (* f(n) is 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}]]
(* A191431 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
(* A191431 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

cp's OEIS Frontend

A191429 Dispersion of ([n*sqrt(2)+2]), where [ ]=floor, by antidiagonals.

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A191432 Dispersion of ([n*x+1/x]), where x=sqrt(2) and [ ]=floor, by antidiagonals.

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PARI

A191436 Dispersion of ([n*x+n+x-1]), where x=(golden ratio) and [ ]=floor, by antidiagonals.

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A191439 Dispersion of ([n*sqrt(2)+n+1/2]), where [ ]=floor, by antidiagonals.

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A191446 Dispersion of [n*sqrt(5)], where [ ]=floor, by antidiagonals.

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Extensions

A191540 Dispersion of (floor(2*n*sqrt(2))), by antidiagonals.

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A191427 Dispersion of ([n*r+3/2]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.

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A191428 Dispersion of ([n*r+r]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.

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Mathematica

A191430 Dispersion of ([n*sqrt(2)+3/2]), where [ ]=floor, by antidiagonals.

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A191540 Dispersion of (floor(2nsqrt(2))), by antidiagonals.