A191442 -id:A191442 - cp's OEIS frontend

A191443 Dispersion of the sequence ([n*sqrt(3)+1]), where [ ]=floor, read by antidiagonals.

1, 2, 3, 4, 6, 5, 7, 11, 9, 8, 13, 20, 16, 14, 10, 23, 35, 28, 25, 18, 12, 40, 61, 49, 44, 32, 21, 15, 70, 106, 85, 77, 56, 37, 26, 17, 122, 184, 148, 134, 97, 65, 46, 30, 19, 212, 319, 257, 233, 169, 113, 80, 52, 33, 22, 368, 553, 446, 404, 293, 196, 139

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....4....7....13
  3....6....11...20...35
  5....9....16...28...49
  8....14...25...44...77
  10...18...32...56...97

Cf. A114537, A035513, A035506, A191442.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;  x = Sqrt[3];
f[n_] := Floor[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}]]
(* A191443 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191443 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191444 Dispersion of ([n*sqrt(3)+3/2]), where [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 11, 8, 10, 7, 20, 15, 18, 13, 9, 36, 27, 32, 24, 17, 12, 63, 48, 56, 43, 30, 22, 14, 110, 84, 98, 75, 53, 39, 25, 16, 192, 146, 171, 131, 93, 69, 44, 29, 19, 334, 254, 297, 228, 162, 121, 77, 51, 34, 21, 580, 441, 515, 396, 282, 211, 134

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....3....6....11...20
  2....4....8....15...27
  5....10...18...32...56
  7....13...24...43...75
  9....17...30...53...93

Cf. A114537, A035513, A035506, A191442.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;  x = Sqr[3];
f[n_] := Floor[n*x+3/2] (* 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}]]
(* A191444 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191444 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191445 Dispersion of ([(n+1)*sqrt(3)]), where [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 6, 5, 4, 12, 10, 8, 7, 22, 19, 15, 13, 9, 39, 34, 27, 24, 17, 11, 69, 60, 48, 43, 31, 20, 14, 121, 105, 84, 76, 55, 36, 25, 16, 211, 183, 147, 133, 96, 64, 45, 29, 18, 367, 318, 256, 232, 168, 112, 79, 51, 32, 21, 637, 552, 445, 403, 292, 195, 138

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...3...6...12..22
  2...5...10..19..34
  4...8...15..27..48
  7...13..24..43..76
  9...17..31..55..96

Cf. A114537, A035513, A035506, A191442.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;  x = Sqr[3];
f[n_] := Floor[n*x+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}]]
(* A191445 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191445 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

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A191443 Dispersion of the sequence ([n*sqrt(3)+1]), where [ ]=floor, read by antidiagonals.

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A191444 Dispersion of ([n*sqrt(3)+3/2]), where [ ]=floor, by antidiagonals.

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Examples

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Programs

Mathematica

A191445 Dispersion of ([(n+1)*sqrt(3)]), where [ ]=floor, by antidiagonals.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica