A191438 -id:A191438 - cp's OEIS frontend

A083087 Square table read by antidiagonals which forms a permutation of the natural numbers: T(n,0) = floor(nx/(x-1))+1, T(n,k+1) = ceiling(xT(n,k)), for n>=0, k>=0, where x = 1 + sqrt(2).

1, 3, 2, 8, 5, 4, 20, 13, 10, 6, 49, 32, 25, 15, 7, 119, 78, 61, 37, 17, 9, 288, 189, 148, 90, 42, 22, 11, 696, 457, 358, 218, 102, 54, 27, 12, 1681, 1104, 865, 527, 247, 131, 66, 29, 14, 4059, 2666, 2089, 1273, 597, 317, 160, 71, 34, 16, 9800, 6437, 5044, 3074

Offset: 0

Paul D. Hanna, Apr 21 2003

The array in A083087 is the dispersion of the sequence given floor(n+1+n*sqrt(2)). The Mathematica program at A191438 generates A083087 using f[n_]:=Floor[n*x+n+1] instead of f[n_]:=Floor[n*x+n]. - Clark Kimberling, Jun 04 2011

			Table begins:
   1  3   8  20  49  119  288 ...
   2  5  13  32  78  189  457 ...
   4 10  25  61 148  358  865 ...
   6 15  37  90 218  527 1273 ...
   7 17  42 102 247  597 1442 ...
   9 22  54 131 317  766 1850 ...
  11 27  66 160 387  935 2258 ...
  12 29  71 172 416 1005 2427 ...
  14 34  83 201 486 1174 2835 ...
  16 39  95 230 556 1343 3243 ...
  18 44 107 259 626 1512 3651 ...
  19 46 112 271 655 1582 3820 ...
  21 51 124 300 725 1751 4228 ...
  23 56 136 329 795 1920 4636 ...
  24 58 141 341 824 1990 4805 ...

Cf. A083088 (first column), A048739 (first row), A083090 (diagonal), A083091 (antidiagonal sums), A083044, A083047, A083050.

z:=10; x:=1+Sqrt(2); S:=[]; for n in [0..z] do for k in [0..n] do if n-k eq 0 then Append(~S, Floor(n*x/(x-1))+1); else Append(~S, Ceiling(x*S[k+1+(n*(n-1) div 2)])); end if; end for; end for; S; // Klaus Brockhaus, Jan 04 2011

Mathematica
```
(See Comments.)
```

T(n,k+1) = 2*T(n,k) + T(n,k-1) + 1 for n>=0, k>=1.

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 *)

A191440 Dispersion of ([n*sqrt(2)+n+3/2]), where [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 8, 6, 4, 20, 15, 11, 5, 49, 37, 28, 13, 7, 119, 90, 69, 32, 18, 9, 288, 218, 168, 78, 44, 23, 10, 696, 527, 407, 189, 107, 57, 25, 12, 1681, 1273, 984, 457, 259, 139, 61, 30, 14, 4059, 3074, 2377, 1104, 626, 337, 148, 73, 35, 16, 9800, 7422, 5740

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....8....20...49
  2....6....15...37...90
  4....11...28...69...168
  5....13...32...78...189
  7....18...44...107..259

Cf. A114537, A035513, A035506, A191438, A191439, A083087.

(* 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+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}]]
(* A191440 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191440 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191441 Dispersion of ([n*x+n+x]), where x=sqrt(2) and [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 4, 2, 12, 7, 3, 31, 19, 9, 5, 77, 48, 24, 14, 6, 188, 118, 60, 36, 16, 8, 456, 287, 147, 89, 41, 21, 10, 1103, 695, 357, 217, 101, 53, 26, 11, 2665, 1680, 864, 526, 246, 130, 65, 28, 13, 6436, 4058, 2088, 1272, 596, 316, 159, 70, 33, 15, 15540, 9799, 5043

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...31...77
  2....7....19...48...118
  3....9....24...60...147
  5....14...36...89...217
  6....16...41...101..246

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+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}]]
(* A191441 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191441 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

cp's OEIS Frontend

A083087 Square table read by antidiagonals which forms a permutation of the natural numbers: T(n,0) = floor(nx/(x-1))+1, T(n,k+1) = ceiling(xT(n,k)), for n>=0, k>=0, where x = 1 + sqrt(2).

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

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Mathematica

A191440 Dispersion of ([n*sqrt(2)+n+3/2]), where [ ]=floor, by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A191441 Dispersion of ([n*x+n+x]), where x=sqrt(2) and [ ]=floor, by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

cp's OEIS Frontend

A083087 Square table read by antidiagonals which forms a permutation of the natural numbers: T(n,0) = floor(n*x/(x-1))+1, T(n,k+1) = ceiling(x*T(n,k)), for n>=0, k>=0, where x = 1 + sqrt(2).

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Magma

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A191440 Dispersion of ([n*sqrt(2)+n+3/2]), where [ ]=floor, by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A191441 Dispersion of ([n*x+n+x]), where x=sqrt(2) and [ ]=floor, by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A083087 Square table read by antidiagonals which forms a permutation of the natural numbers: T(n,0) = floor(nx/(x-1))+1, T(n,k+1) = ceiling(xT(n,k)), for n>=0, k>=0, where x = 1 + sqrt(2).