A191426 -id:A191426 - cp's OEIS frontend

A191452 Dispersion of (4,8,12,16,...), by antidiagonals.

1, 4, 2, 16, 8, 3, 64, 32, 12, 5, 256, 128, 48, 20, 6, 1024, 512, 192, 80, 24, 7, 4096, 2048, 768, 320, 96, 28, 9, 16384, 8192, 3072, 1280, 384, 112, 36, 10, 65536, 32768, 12288, 5120, 1536, 448, 144, 40, 11, 262144, 131072, 49152, 20480, 6144, 1792, 576

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...4....16...64....256
  2...8....32...128...512
  3...12...48...192...768
  5...20...80...320...1280
  6...24...96...384...1536

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A114537, A035513, A035506, A191449.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=4n  (* 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}]]
(* A191452 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191452 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A254105 Dispersion of A055938; starting from its complementary sequence A005187 as the first column of square array A(row,col), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

1, 2, 3, 5, 6, 4, 12, 13, 9, 7, 27, 28, 20, 14, 8, 58, 59, 43, 29, 17, 10, 121, 122, 90, 60, 36, 21, 11, 248, 249, 185, 123, 75, 44, 24, 15, 503, 504, 376, 250, 154, 91, 51, 30, 16, 1014, 1015, 759, 505, 313, 186, 106, 61, 33, 18, 2037, 2038, 1526, 1016, 632, 377, 217, 124, 68, 37, 19, 4084, 4085, 3061, 2039, 1271, 760, 440, 251, 139, 76, 40, 22

Offset: 1

Antti Karttunen, Jan 26 2015

This sequence is one instance of Clark Kimberling's generic dispersion arrays. Paraphrasing his explanation in A191450, mutatis mutandis, we have the following definition:
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) = {index of the row of D that contains n} is a fractal sequence. In this case s(n) = A055938(n), t(n) = A005187(n) [from term A005187(1) onward] and u(n) = A254112(n).
For other examples of such sequences, see the Crossrefs section. For a general introduction, please follow the Kimberling references.
The main diagonal: 1, 6, 20, 60, 154, 377, 887, 2040, 4598, 10229, 22515, 49139, ...

			The top left corner of the array:
   1,  2,  5,  12,  27,  58,  121,  248,  503,  1014,  2037,  4084
   3,  6, 13,  28,  59, 122,  249,  504, 1015,  2038,  4085,  8180
   4,  9, 20,  43,  90, 185,  376,  759, 1526,  3061,  6132, 12275
   7, 14, 29,  60, 123, 250,  505, 1016, 2039,  4086,  8181, 16372
   8, 17, 36,  75, 154, 313,  632, 1271, 2550,  5109, 10228, 20467
  10, 21, 44,  91, 186, 377,  760, 1527, 3062,  6133, 12276, 24563
  11, 24, 51, 106, 217, 440,  887, 1782, 3573,  7156, 14323, 28658
  15, 30, 61, 124, 251, 506, 1017, 2040, 4087,  8182, 16373, 32756
  16, 33, 68, 139, 282, 569, 1144, 2295, 4598,  9205, 18420, 36851
  18, 37, 76, 155, 314, 633, 1272, 2551, 5110, 10229, 20468, 40947
etc.

Antti Karttunen, Table of n, a(n) for n = 1..120; the first 15 antidiagonals of array
Clark Kimberling, Interspersions and Dispersions.
Clark Kimberling, Interspersions and Dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Index entries for sequences that are permutations of the natural numbers

Inverse: A254106.
Transpose: A254107.
Column 1: A005187.
Cf. also A000325, A095768, A123720 (Seem to be rows 1 - 3, the last one from its second term onward.)
Columnd index of n: A254111, Row index: A254112.
Cf. A002260, A004736, A055938, A233275-A233278.
Examples of other arrays of dispersions: A114537, A035513, A035506, A191449, A191450, A191426-A191455.

(define (A254105 n) (A254105bi (A002260 n) (A004736 n)))
(define (A254105bi row col) (if (= 1 col) (A005187 row) (A055938 (A254105bi row (- col 1)))))

If col = 1, then A(row,col) = A005187(row), otherwise A(row,col) = A055938(A(row,col-1)).

A191668 Dispersion of A016825 (4k+2, k>0), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 6, 10, 4, 22, 38, 14, 5, 86, 150, 54, 18, 7, 342, 598, 214, 70, 26, 8, 1366, 2390, 854, 278, 102, 30, 9, 5462, 9558, 3414, 1110, 406, 118, 34, 11, 21846, 38230, 13654, 4438, 1622, 470, 134, 42, 12, 87382, 152918, 54614, 17750, 6486, 1878, 534, 166

Offset: 1

Clark Kimberling, Jun 11 2011

For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The six sequences and dispersions are listed here:
...
A191452=dispersion of A008586 (4k, k>=1)
A191667=dispersion of A016813 (4k+1, k>=1)
A191668=dispersion of A016825 (4k+2, k>=0)
A191669=dispersion of A004767 (4k+3, k>=0)
A191670=dispersion of A042968 (1 or 2 or 3 mod 4 and >=2)
A191671=dispersion of A004772 (0 or 1 or 3 mod 4 and >=2)
A191672=dispersion of A004773 (0 or 1 or 2 mod 4 and >=2)
A191673=dispersion of A004773 (0 or 2 or 3 mod 4 and >=2)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191452 has 1st col A042968, all else A008486
A191667 has 1st col A004772, all else A016813
A191668 has 1st col A042965, all else A016825
A191669 has 1st col A004773, all else A004767
A191670 has 1st col A008486, all else A042968
A191671 has 1st col A016813, all else A004772
A191672 has 1st col A016825, all else A042965
A191673 has 1st col A004767, all else A004773
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 3), then (a,b,c,a,b,c,a,b,c,...) is given by
   a*f(n+2)+b*f(n+1)+c*f(n), so that "(a or b or c mod m)" is given by
   a*f(n+2)+b*f(n+1)+c*f(n)+m*floor((n-1)/3)), for n>=1.

			Northwest corner:
.    1   2    6   22    86    342   1366    5462   21846    87382
.    3  10   38  150   598   2390   9558   38230  152918   611670
.    4  14   54  214   854   3414  13654   54614  218454   873814
.    5  18   70  278  1110   4438  17750   70998  283990  1135958
.    7  26  102  406  1622   6486  25942  103766  415062  1660246
.    8  30  118  470  1878   7510  30038  120150  480598  1922390
.    9  34  134  534  2134   8534  34134  136534  546134  2184534
.   11  42  166  662  2646  10582  42326  169302  677206  2708822
.   12  46  182  726  2902  11606  46422  185686  742742  2970966
.   13  50  198  790  3158  12630  50518  202070  808278  3233110

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Row 1: A047849.

Cf. A000302, A042965, A016825, A191672, A191426.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
f[n_] := 4*n-2
Table[f[n], {n, 1, 30}]  (* A016825 *)
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}]] (* A191668 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191668 *)
(* Conjectured: *) Grid[Table[(8 + (3*Floor[(4*n + 1)/3] - 2)*4^k)/12, {n, 10}, {k, 10}]] (* L. Edson Jeffery, Feb 14 2015 *)

Conjecture: a(n,k) = (8 + (3*floor((4*n + 1)/3) - 2)*4^k)/12 = (8 + (3*A042965(n+1) - 2)*A000302(k))/12. - L. Edson Jeffery, Feb 14 2015

A191669 Dispersion of A004767 (4k+3, k>=0), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 11, 7, 4, 43, 27, 15, 5, 171, 107, 59, 19, 6, 683, 427, 235, 75, 23, 8, 2731, 1707, 939, 299, 91, 31, 9, 10923, 6827, 3755, 1195, 363, 123, 35, 10, 43691, 27307, 15019, 4779, 1451, 491, 139, 39, 12, 174763, 109227, 60075, 19115, 5803, 1963, 555, 155

Offset: 1

Clark Kimberling, Jun 11 2011

For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The six sequences and dispersions are listed here:
...
A191452=dispersion of A008586 (4k, k>=1)
A191667=dispersion of A016813 (4k+1, k>=1)
A191668=dispersion of A016825 (4k+2, k>=0)
A191669=dispersion of A004767 (4k+3, k>=0)
A191670=dispersion of A042968 (1 or 2 or 3 mod 4 and >=2)
A191671=dispersion of A004772 (0 or 1 or 3 mod 4 and >=2)
A191672=dispersion of A004773 (0 or 1 or 2 mod 4 and >=2)
A191673=dispersion of A004773 (0 or 2 or 3 mod 4 and >=2)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191452 has 1st col A042968, all else A008486
A191667 has 1st col A004772, all else A016813
A191668 has 1st col A042965, all else A016825
A191669 has 1st col A004773, all else A004767
A191670 has 1st col A008486, all else A042968
A191671 has 1st col A016813, all else A004772
A191672 has 1st col A016825, all else A042965
A191673 has 1st col A004767, all else A004773
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 3), then (a,b,c,a,b,c,a,b,c,...) is given by
   a*f(n+2)+b*f(n+1)+c*f(n), so that "(a or b or c mod m)" is given by
   a*f(n+2)+b*f(n+1)+c*f(n)+m*floor((n-1)/3)), for n>=1.

			Northwest corner:
1...3....11....43....171
2...7....27....107...427
4...15...59....235...939
5...19...75....299...1195
6...23...91....363...1451

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Row 1: A007583, Row 2: A136412.

Cf. A004773, A004767, A191673, A191426.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
f[n_] := 4*n-1
Table[f[n], {n, 1, 30}] (* A004767 *)
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}]] (* A191669 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191669 *)

A035507 Inverse Stolarsky array read by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 7, 12, 5, 9, 20, 33, 6, 14, 25, 54, 88, 8, 17, 38, 67, 143, 232, 10, 22, 46, 101, 177, 376, 609, 11, 27, 59, 122, 266, 465, 986, 1596, 13, 30, 72, 156, 321, 698, 1219, 2583, 4180, 15, 35, 80, 190, 410, 842, 1829, 3193, 6764, 10945, 16, 41, 93, 211, 499

Offset: 0

N. J. A. Sloane

The inverse Stolarky array is the dispersion of the sequence u given by u(n) = floor(n*x + x + n + 1 - x/2), where x=(golden ratio). For a discussion of dispersions, see A191426.

			Top left hand corner of array:
  1,    4,   12,   33,   88,  232, ...
  2,    7,   20,   54,  143,  376, ...
  3,    9,   25,   67,  177,  465, ...
  5,   14,   38,  101,  266,  698, ...
  6,   17,   46,  122,  321,  842, ...

C. Kimberling, Interspersions
C. Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society 117 (1993) 313-321.
N. J. A. Sloane, Classic Sequences

Cf. A035506 (Stolarsky array), A191426.

with(combinat, fibonacci): gold:=(1+sqrt(5))/2: c1:=n->piecewise(n<>1,round((n-1)*gold),1): c2:=n->c1(n)+floor((2*c1(n)+1)*gold/2)+1: inv_stol:=(n,k)->fibonacci(2*k-3)-1-c1(n)*fibonacci(2*k-4)+c2(n)*fibonacci(2*k-2): seq(seq(inv_stol(n+1-k,k),k=1..n),n=1..11); inv_stol2:=(n,k)->(1+c0(n))*fibonacci(2*k-3)+(1+floor((2*c0(n)+1)*gold/2))*fibonacci(2*k-2)-1:seq(seq(inv_stol2(n+1-k,k),k=1..n),n=1..11);  # C. Ronaldo, Dec 31 2004

(* program generates the dispersion array T of the complement 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 + n + 1 - 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}]]
(* Inverse Stolarsky array, A035507 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
(* array as a sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

The term in row n and column k of the inverse Stolarsky array has the following expression: a(n, k) = F(2k-3) - 1 - c1(n)*F(2k-4) + c2(n)*F(2k-2), where F is the Fibonacci sequence; c1(n)=1 if n=1, [(n-1)*tau] if n>1 (first column of the Inverse Stolarsky array) and c2(n) = c1(n) + 1 + floor((2*c1(n)+1)*tau/2) (second column of the Inverse Stolarsky array). tau = (1+sqrt(5))/2 and [] denotes the nearest integer function. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 31 2004

Also, the following recurrence holds: a(n, k) = 3*a(n, k-1) - a(n, k-2) + 1 with a(n, 1)=c1(n) and a(n, 2)=c2(n). - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 31 2004

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 31 2004

Mathematica program, extended example, and comments from Clark Kimberling, Jun 03 2011

A191449 Dispersion of (3,6,9,12,15,...), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 9, 6, 4, 27, 18, 12, 5, 81, 54, 36, 15, 7, 243, 162, 108, 45, 21, 8, 729, 486, 324, 135, 63, 24, 10, 2187, 1458, 972, 405, 189, 72, 30, 11, 6561, 4374, 2916, 1215, 567, 216, 90, 33, 13, 19683, 13122, 8748, 3645, 1701, 648, 270, 99, 39, 14, 59049

Offset: 1

Clark Kimberling, Jun 05 2011

Transpose of A141396.
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....9....27...81
  2...6....18...54...162
  4...12...36...108..324
  5...15...45...135..405
  7...21...63...189..567

Cf. A114537, A035513, A035506,
A054582: dispersion of (2,4,6,8,...).
A191450: dispersion of (2,5,8,11,...).
A191451: dispersion of (4,7,10,13,...).
A191452: dispersion of (4,8,12,16,...).

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=3n (* 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}]]
(* A191449 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191449 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

T(i,j)=T(i,1)*T(1,j)=floor((3i-1)/2)*3^(j-1).

A191671 Dispersion of A004772 (>1 and congruent to 0 or 2 or 3 mod 4), by antidiagonals.

Original entry on oeis.org

1, 2, 5, 3, 7, 9, 4, 10, 12, 13, 6, 14, 16, 18, 17, 8, 19, 22, 24, 23, 21, 11, 26, 30, 32, 31, 28, 25, 15, 35, 40, 43, 42, 38, 34, 29, 20, 47, 54, 58, 56, 51, 46, 39, 33, 27, 63, 72, 78, 75, 68, 62, 52, 44, 37, 36, 84, 96, 104, 100, 91, 83, 70, 59, 50, 41

Offset: 1

Clark Kimberling, Jun 11 2011

For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The six sequences and dispersions are listed here:
...
A191452=dispersion of A008586 (4k, k>=1)
A191667=dispersion of A016813 (4k+1, k>=1)
A191668=dispersion of A016825 (4k+2, k>=0)
A191669=dispersion of A004767 (4k+3, k>=0)
A191670=dispersion of A042968 (1 or 2 or 3 mod 4 and >=2)
A191671=dispersion of A004772 (0 or 1 or 3 mod 4 and >=2)
A191672=dispersion of A004773 (0 or 1 or 2 mod 4 and >=2)
A191673=dispersion of A004773 (0 or 2 or 3 mod 4 and >=2)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191452 has 1st col A042968, all else A008486
A191667 has 1st col A004772, all else A016813
A191668 has 1st col A042965, all else A016825
A191669 has 1st col A004773, all else A004767
A191670 has 1st col A008486, all else A042968
A191671 has 1st col A016813, all else A004772
A191672 has 1st col A016825, all else A042965
A191673 has 1st col A004767, all else A004773
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 3), then (a,b,c,a,b,c,a,b,c,...) is given by
   a*f(n+2)+b*f(n+1)+c*f(n), so that "(a or b or c mod m)" is given by
   a*f(n+2)+b*f(n+1)+c*f(n)+m*floor((n-1)/3)), for n>=1.

			Northwest corner:
1....2....3....4....6
5....7....10...14...19
9....12...16...22...30
13...18...24...32...43
17...23...31...42...56

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A016813, A004772, A191667, A191452.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a = 2; b = 3; c2 = 4; m[n_] := If[Mod[n, 3] == 0, 1, 0];
f[n_] := a*m[n + 2] + b*m[n + 1] + c2*m[n] + 4*Floor[(n - 1)/3]
Table[f[n], {n, 1, 30}]  (* A004772 *)
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}]] (* A191671 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191671  *)

A191672 Dispersion of A042965 (>1 and congruent to 0 or 1 or 3 mod 4), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 5, 4, 6, 8, 7, 9, 10, 12, 11, 13, 15, 14, 17, 16, 19, 21, 20, 18, 24, 23, 27, 29, 28, 25, 22, 33, 32, 37, 40, 39, 35, 31, 26, 45, 44, 51, 55, 53, 48, 43, 36, 30, 61, 60, 69, 75, 72, 65, 59, 49, 41, 34, 83, 81, 93, 101, 97, 88, 80, 67, 56, 47, 38

Offset: 1

Clark Kimberling, Jun 11 2011

For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The six sequences and dispersions are listed here:
...
A191452=dispersion of A008586 (4k, k>=1)
A191667=dispersion of A016813 (4k+1, k>=1)
A191668=dispersion of A016825 (4k+2, k>=0)
A191669=dispersion of A004767 (4k+3, k>=0)
A191670=dispersion of A042968 (1 or 2 or 3 mod 4 and >=2)
A191671=dispersion of A004772 (0 or 1 or 3 mod 4 and >=2)
A191672=dispersion of A004773 (0 or 1 or 2 mod 4 and >=2)
A191673=dispersion of A004773 (0 or 2 or 3 mod 4 and >=2)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191452 has 1st col A042968, all else A008486
A191667 has 1st col A004772, all else A016813
A191668 has 1st col A042965, all else A016825
A191669 has 1st col A004773, all else A004767
A191670 has 1st col A008486, all else A042968
A191671 has 1st col A016813, all else A004772
A191672 has 1st col A016825, all else A042965
A191673 has 1st col A004767, all else A004773
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 3), then (a,b,c,a,b,c,a,b,c,...) is given by
   a*f(n+2)+b*f(n+1)+c*f(n), so that "(a or b or c mod m)" is given by
   a*f(n+2)+b*f(n+1)+c*f(n)+m*floor((n-1)/3)), for n>=1.

			Northwest corner:
1....3...5....8....12
2....4...7....11...16
6....9...13...19...27
10...15..21...29...40
14...20..28...39...53

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A016825, A042965, A191668, A191452.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
a = 3; b = 4; c2 = 5; m[n_] := If[Mod[n, 3] == 0, 1, 0];
f[n_] := a*m[n + 2] + b*m[n + 1] + c2*m[n] + 4*Floor[(n - 1)/3]
Table[f[n], {n, 1, 30}] (* A042965 *)
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}]] (* A191672 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191672 *)

A191448 Dispersion of the odd integers greater than 1, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 7, 5, 4, 15, 11, 9, 6, 31, 23, 19, 13, 8, 63, 47, 39, 27, 17, 10, 127, 95, 79, 55, 35, 21, 12, 255, 191, 159, 111, 71, 43, 25, 14, 511, 383, 319, 223, 143, 87, 51, 29, 16, 1023, 767, 639, 447, 287, 175, 103, 59, 33, 18, 2047, 1535, 1279, 895, 575

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...3...7...15..31
  2...5...11..23..47
  4...9...19..39..79
  6...13..27..55..111
  8...17..35..71..143

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

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_] :=2n+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}]]
(* A191448 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191448 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191545 Dispersion of (floor(9*n/4)), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 9, 13, 11, 7, 20, 29, 24, 15, 8, 45, 65, 54, 33, 18, 10, 101, 146, 121, 74, 40, 22, 12, 227, 328, 272, 166, 90, 49, 27, 14, 510, 738, 612, 373, 202, 110, 60, 31, 16, 1147, 1660, 1377, 839, 454, 247, 135, 69, 36, 17, 2580, 3735, 3098, 1887

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    20
  3   6    13   29   65
  5   11   25   54   121
  7   15   33   74   166
  8   18   40   90   202

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[9n/4]   (* 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}]]
(* A191545 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191545 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

cp's OEIS Frontend

A191452 Dispersion of (4,8,12,16,...), by antidiagonals.

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A254105 Dispersion of A055938; starting from its complementary sequence A005187 as the first column of square array A(row,col), read by antidiagonals A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A191668 Dispersion of A016825 (4k+2, k>0), by antidiagonals.

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A191669 Dispersion of A004767 (4k+3, k>=0), by antidiagonals.

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A035507 Inverse Stolarsky array read by antidiagonals.

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A191449 Dispersion of (3,6,9,12,15,...), by antidiagonals.

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A191671 Dispersion of A004772 (>1 and congruent to 0 or 2 or 3 mod 4), by antidiagonals.

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A191672 Dispersion of A042965 (>1 and congruent to 0 or 1 or 3 mod 4), by antidiagonals.

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