A191545 -id:A191545 - cp's OEIS frontend

A191536 Dispersion of (3+floor(n*sqrt(2))), by antidiagonals.

1, 4, 2, 8, 5, 3, 14, 10, 7, 6, 22, 17, 12, 11, 9, 34, 27, 19, 18, 15, 13, 51, 41, 29, 28, 24, 21, 16, 75, 60, 44, 42, 36, 32, 25, 20, 109, 87, 65, 62, 53, 48, 38, 31, 23, 157, 126, 94, 90, 77, 70, 56, 46, 35, 26, 225, 181, 135, 130, 111, 101, 82, 68, 52, 39

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...4....8....14...22
  2...5....10...17...27
  3...7....12...19...29
  6...11...18...28...42
  9...15...24...36...54

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_] :=3+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}]]
(* A191536 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191536 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

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 *)

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 *)

cp's OEIS Frontend

A191536 Dispersion of (3+floor(n*sqrt(2))), by antidiagonals.

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A191540 Dispersion of (floor(2*n*sqrt(2))), 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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Mathematica

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

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Mathematica

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