cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A191667 Dispersion of A016813 (4k+1, k>1), by antidiagonals.

Original entry on oeis.org

1, 5, 2, 21, 9, 3, 85, 37, 13, 4, 341, 149, 53, 17, 6, 1365, 597, 213, 69, 25, 7, 5461, 2389, 853, 277, 101, 29, 8, 21845, 9557, 3413, 1109, 405, 117, 33, 10, 87381, 38229, 13653, 4437, 1621, 469, 133, 41, 11, 349525, 152917, 54613, 17749, 6485, 1877, 533
Offset: 1

Views

Author

Clark Kimberling, Jun 11 2011

Keywords

Comments

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.

Examples

			Northwest corner:
1....5....21....85....341
2....9....37....149...597
3....13...53....213...853
4....17...69....277...1109
6....25...101...405...1621

Links

Crossrefs

Row 1: A002450.
Cf. A004772, A016813, A191671, A191426.

Programs

  • Mathematica
    (* 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}]  (* A016813 *)
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}]] (* A191667 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191667  *)

A191670 Dispersion of A042968 (>1 and congruent to 1 or 2 or 3 mod 4), by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 5, 9, 11, 12, 7, 13, 15, 17, 16, 10, 18, 21, 23, 22, 20, 14, 25, 29, 31, 30, 27, 24, 19, 34, 39, 42, 41, 37, 33, 28, 26, 46, 53, 57, 55, 50, 45, 38, 32, 35, 62, 71, 77, 74, 67, 61, 51, 43, 36, 47, 83, 95, 103, 99, 90, 82, 69, 58, 49, 40, 63
Offset: 1

Views

Author

Clark Kimberling, Jun 11 2011

Keywords

Comments

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.

Examples

			Northwest corner:
1....2....3....5....7
4....6....9....13...18
8....11...15...21...29
12...17...23...31...42
16...22...30...41...55

Links

Crossrefs

Row 1: A155167, Row 2: A171861.
Cf. A008486, A042968, A191673, A191452.

Programs

  • Mathematica
    (* 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 = 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}] (* A042968 *)
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}]] (* A191670 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191670 *)

A191673 Dispersion of A004773 (>1 and congruent to 0 or 1 or 2 mod 4), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 8, 10, 11, 9, 12, 14, 16, 15, 13, 17, 20, 22, 21, 19, 18, 24, 28, 30, 29, 26, 23, 25, 33, 38, 41, 40, 36, 32, 27, 34, 45, 52, 56, 54, 49, 44, 37, 31, 46, 61, 70, 76, 73, 66, 60, 50, 42, 35, 62, 82, 94, 102, 98, 89, 81, 68, 57, 48, 39, 84
Offset: 1

Views

Author

Clark Kimberling, Jun 11 2011

Keywords

Comments

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.

Examples

			Northwest corner:
1....2....4....6....9
3....5....8....12...17
7....10...14...20...28
11...16...22...30...41
15...21...29...40...54

Links

Crossrefs

Cf. A004767, A004773, A191669, A191452.

Programs

  • Mathematica
    (* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
a = 2; 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}] (* A004773 *)
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}]] (* A191673 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191673 *)

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

Views

Author

Clark Kimberling, Jun 11 2011

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

Row 1: A047849.
Cf. A000302, A042965, A016825, A191672, A191426.

Programs

  • Mathematica
    (* 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 *)

Formula

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

Views

Author

Clark Kimberling, Jun 11 2011

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

Row 1: A007583, Row 2: A136412.
Cf. A004773, A004767, A191673, A191426.

Programs

  • Mathematica
    (* 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 *)

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

Views

Author

Clark Kimberling, Jun 05 2011

Keywords

Comments

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.

Examples

			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

Crossrefs

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,...).

Programs

  • Mathematica
    (* 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 *)

Formula

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

Views

Author

Clark Kimberling, Jun 11 2011

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

Cf. A016813, A004772, A191667, A191452.

Programs

  • Mathematica
    (* 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

Views

Author

Clark Kimberling, Jun 11 2011

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

Cf. A016825, A042965, A191668, A191452.

Programs

  • Mathematica
    (* 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 *)
Showing 1-8 of 8 results.

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