A133299 -id:A133299 - cp's OEIS frontend

A191453 Dispersion of (2floor(3n/2)), by antidiagonals.

1, 2, 3, 6, 8, 4, 18, 24, 12, 5, 54, 72, 36, 14, 7, 162, 216, 108, 42, 20, 9, 486, 648, 324, 126, 60, 26, 10, 1458, 1944, 972, 378, 180, 78, 30, 11, 4374, 5832, 2916, 1134, 540, 234, 90, 32, 13, 13122, 17496, 8748, 3402, 1620, 702, 270, 96, 38, 15, 39366

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....6....18...54
  3...8....24...72...216
  4...12...36...108..324
  5...14...42...126..378
  7...20...60...180..540

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=2Floor[3n/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}]]
(* A191453 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191453 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191454 Dispersion of (2floor(nr)), where r=(golden ratio), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 6, 8, 4, 18, 24, 12, 5, 58, 76, 38, 16, 7, 186, 244, 122, 50, 22, 9, 600, 788, 394, 160, 70, 28, 10, 1940, 2550, 1274, 516, 226, 90, 32, 11, 6276, 8250, 4122, 1668, 730, 290, 102, 34, 13, 20308, 26696, 13338, 5396, 2362, 938, 330, 110, 42, 14, 65718

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....6....18...58
  3...8....24...76...244
  4...12...38...122..394
  5...16...50...160..516
  7...22...70...226..730

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_] :=2Floor[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}]]
(* A191454 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191454 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191537 Dispersion of (4n-floor(nsqrt(2))), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 8, 6, 4, 21, 16, 11, 5, 55, 42, 29, 13, 7, 143, 109, 75, 34, 19, 9, 370, 282, 194, 88, 50, 24, 10, 957, 730, 502, 228, 130, 63, 26, 12, 2475, 1888, 1299, 590, 337, 163, 68, 32, 14, 6400, 4882, 3359, 1526, 872, 422, 176, 83, 37, 15, 16550, 12624

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,  3,  8,  21,  55, ...
  2,  6, 16,  42, 109, ...
  4, 11, 29,  75, 194, ...
  5, 13, 34,  88, 228, ...
  7, 19, 50, 130, 337, ...

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_] :=4n-Floor[n*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}]]  (* A191537 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191537 sequence *)
(* Clark Kimberling, Jun 06 2011 *)

A191538 Dispersion of (4n-floor(nsqrt(3))), by antidiagonals.

Original entry on oeis.org

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

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,  3,  7,  16,  37, ...
  2,  5, 12,  28,  64, ...
  4, 10, 23,  53, 121, ...
  6, 14, 32,  73, 166, ...
  8, 19, 44, 100, 227, ...

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_] :=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 *)

A191539 Dispersion of (5n-floor(nsqrt(5))), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 9, 6, 4, 25, 17, 12, 5, 70, 47, 34, 14, 7, 194, 130, 94, 39, 20, 8, 537, 360, 260, 108, 56, 23, 10, 1485, 996, 719, 299, 155, 64, 28, 11, 4105, 2753, 1988, 827, 429, 177, 78, 31, 13, 11346, 7610, 5495, 2286, 1186, 490, 216, 86, 36, 15, 31360, 21034

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...3....9....25...70
  2...6....17...47...130
  4...12...34...94...260
  5...14...39...108..299
  7...20...56...155..429

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_] :=5n-Floor[n*Sqrt[5]]   (* 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}]]  (* A191539 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191539 sequence *)

A191541 Dispersion of (2floor(nsqrt(2))), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 8, 5, 10, 22, 14, 6, 28, 62, 38, 16, 7, 78, 174, 106, 44, 18, 9, 220, 492, 298, 124, 50, 24, 11, 622, 1390, 842, 350, 140, 66, 30, 12, 1758, 3930, 2380, 988, 394, 186, 84, 32, 13, 4972, 11114, 6730, 2794, 1114, 526, 236, 90, 36, 15, 14062, 31434

Offset: 1

Clark Kimberling, Jun 07 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....10...28
  3...8....22...62...174
  5...14...38...106..298
  6...16...44...124..350
  7...18...50...140..394

Cf. A114537, A035513, A035506, A191540.

(* Program generates the dispersion array T of the complement of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] :=2*Floor[n*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, 10}, {j, 1, 10}]]
(* A191541 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191541 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191542 Dispersion of (2floor(nsqrt(3))), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 6, 10, 4, 20, 34, 12, 5, 68, 116, 40, 16, 7, 234, 400, 138, 54, 24, 8, 810, 1384, 478, 186, 82, 26, 9, 2804, 4794, 1654, 644, 284, 90, 30, 11, 9712, 16606, 5728, 2230, 982, 310, 102, 38, 13, 33642, 57524, 19842, 7724, 3400, 1072, 352, 130, 44, 14

Offset: 1

Clark Kimberling, Jun 07 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....6....20...68
  3...10...34...116..400
  4...12...40...138..478
  5...16...54...186..644
  7...24...82...284..982

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of the complement of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] :=2*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, 10}, {j, 1, 10}]]
(* A191542 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191542 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191543 Dispersion of (floor(8*n/3)), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 8, 4, 13, 21, 10, 6, 34, 56, 26, 16, 7, 90, 149, 69, 42, 18, 9, 240, 397, 184, 112, 48, 24, 11, 640, 1058, 490, 298, 128, 64, 29, 12, 1706, 2821, 1306, 794, 341, 170, 77, 32, 14, 4549, 7522, 3482, 2117, 909, 453, 205, 85, 37, 15, 12130, 20058

Offset: 1

Clark Kimberling, Jun 07 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    13   34
  3   8    21   56   149
  4   10   26   69   184
  6   16   42   112  298
  7   18   48   128  341

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of the complement of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] := Floor[8n/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, 10}, {j, 1, 10}]]
(* A191543 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191543 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191544 Dispersion of (floor(7*n/3)), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 9, 16, 11, 6, 21, 37, 25, 14, 8, 49, 86, 58, 32, 18, 10, 114, 200, 135, 74, 42, 23, 12, 266, 466, 315, 172, 98, 53, 28, 13, 620, 1087, 735, 401, 228, 123, 65, 30, 15, 1446, 2536, 1715, 935, 532, 287, 151, 70, 35, 17, 3374, 5917, 4001, 2181

Offset: 1

Clark Kimberling, Jun 09 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    9   21
  3   7    16   37  86
  5   11   25   58  135
  6   14   32   74  172
  8   18   42   98  228

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of the complement of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] := Floor[7n/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, 10}, {j, 1, 10}]]
(* A191544 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191544 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

cp's OEIS Frontend

A191453 Dispersion of (2*floor(3*n/2)), by antidiagonals.

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A191454 Dispersion of (2*floor(n*r)), where r=(golden ratio), by antidiagonals.

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A191537 Dispersion of (4*n-floor(n*sqrt(2))), by antidiagonals.

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A191538 Dispersion of (4*n-floor(n*sqrt(3))), by antidiagonals.

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A191539 Dispersion of (5*n-floor(n*sqrt(5))), by antidiagonals.

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A191541 Dispersion of (2*floor(n*sqrt(2))), by antidiagonals.

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A191542 Dispersion of (2*floor(n*sqrt(3))), by antidiagonals.

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A191543 Dispersion of (floor(8*n/3)), by antidiagonals.

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A191544 Dispersion of (floor(7*n/3)), by antidiagonals.

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A191453 Dispersion of (2floor(3n/2)), by antidiagonals.

A191454 Dispersion of (2floor(nr)), where r=(golden ratio), by antidiagonals.

A191537 Dispersion of (4n-floor(nsqrt(2))), by antidiagonals.

A191538 Dispersion of (4n-floor(nsqrt(3))), by antidiagonals.

A191539 Dispersion of (5n-floor(nsqrt(5))), by antidiagonals.

A191541 Dispersion of (2floor(nsqrt(2))), by antidiagonals.

A191542 Dispersion of (2floor(nsqrt(3))), by antidiagonals.