A191667 -id:A191667 - cp's OEIS frontend

A191702 Dispersion of A008587 (5,10,15,20,25,30,...), by antidiagonals.

1, 5, 2, 25, 10, 3, 125, 50, 15, 4, 625, 250, 75, 20, 6, 3125, 1250, 375, 100, 30, 7, 15625, 6250, 1875, 500, 150, 35, 8, 78125, 31250, 9375, 2500, 750, 175, 40, 9, 390625, 156250, 46875, 12500, 3750, 875, 200, 45, 11, 1953125, 781250, 234375, 62500, 18750

Offset: 1

Clark Kimberling, Jun 12 2011

For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type
"(a or b or c or d mod m)", used in Mathematica programs for A191707-A191711):  if f(n)=(n mod 3), then
  (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by
   a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n), so that for n>=1,
"(a, b, c, d mod m)" is given by
   a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

			Northwest corner:
  1...5....25....125...625
  2...10...50....250...1250
  3...15...75....375...1875
  4...20...100...500...2500
  6...30...150...750...3750

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

Cf. A047201, A008587, A191707, A191426.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
f[n_] := 5n
Table[f[n], {n, 1, 30}]  (* A008587 *)
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}]]
(* A191702 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191702  *)

T(i,j) = T(i,1)*T(1,j) = (i-1+floor((i+3)/4))*5^(j-1), i>=1, j>=1.

A191722 Dispersion of A008851, (numbers >1 and congruent to 0 or 1 mod 5), by antidiagonals.

Original entry on oeis.org

1, 5, 2, 15, 6, 3, 40, 16, 10, 4, 101, 41, 26, 11, 7, 255, 105, 66, 30, 20, 8, 640, 265, 166, 76, 51, 21, 9, 1601, 665, 416, 191, 130, 55, 25, 12, 4005, 1665, 1041, 480, 326, 140, 65, 31, 13, 10015, 4165, 2605, 1201, 816, 351, 165, 80, 35, 14, 25040, 10415

Offset: 1

Clark Kimberling, Jun 13 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
...
A191722=dispersion of A008851 (0, 1 mod 5 and >1)
A191723=dispersion of A047215 (0, 2 mod 5 and >1)
A191724=dispersion of A047218 (0, 3 mod 5 and >1)
A191725=dispersion of A047208 (0, 4 mod 5 and >1)
A191726=dispersion of A047216 (1, 2 mod 5 and >1)
A191727=dispersion of A047219 (1, 3 mod 5 and >1)
A191728=dispersion of A047209 (1, 4 mod 5 and >1)
A191729=dispersion of A047221 (2, 3 mod 5 and >1)
A191730=dispersion of A047211 (2, 4 mod 5 and >1)
A191731=dispersion of A047204 (3, 4 mod 5 and >1)
...
A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191722 has 1st col A047202, all else A008851
A191723 has 1st col A047206, all else A047215
A191724 has 1st col A032793, all else A047218
A191725 has 1st col A047223, all else A047208
A191726 has 1st col A047205, all else A047216
A191727 has 1st col A047212, all else A047219
A191728 has 1st col A047222, all else A047209
A191729 has 1st col A008854, all else A047221
A191730 has 1st col A047220, all else A047211
A191731 has 1st col A047217, all else A047204
...
A191732 has 1st col A000851, all else A047202
A191733 has 1st col A047215, all else A047206
A191734 has 1st col A047218, all else A032793
A191735 has 1st col A047208, all else A047223
A191736 has 1st col A047216, all else A047205
A191737 has 1st col A047219, all else A047212
A191738 has 1st col A047209, all else A047222
A191739 has 1st col A047221, all else A008854
A191740 has 1st col A047211, all else A047220
A191741 has 1st col A047204, all else A047217
...
Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

			Northwest corner:
1....5....15...40...101
2....6....16...41...105
3....10...26...66...166
4....11...30...76...191
7....20...51...130..326
8....21...55...140..351

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A047202, A008851, A191732, A191702, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=5; b=6; m[n_]:=If[Mod[n,2]==0,1,0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}]  (* A008851 *)
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}]]
(* A191722 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191722  *)

A191723 Dispersion of A047215, (numbers >1 and congruent to 0 or 2 mod 5), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 7, 4, 12, 17, 10, 6, 30, 42, 25, 15, 8, 75, 105, 62, 37, 20, 9, 187, 262, 155, 92, 50, 22, 11, 467, 655, 387, 230, 125, 55, 27, 13, 1167, 1637, 967, 575, 312, 137, 67, 32, 14, 2917, 4092, 2417, 1437, 780, 342, 167, 80, 35, 16, 7292, 10230, 6042

Offset: 1

Clark Kimberling, Jun 13 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
...
A191722=dispersion of A008851 (0, 1 mod 5 and >1)
A191723=dispersion of A047215 (0, 2 mod 5 and >1)
A191724=dispersion of A047218 (0, 3 mod 5 and >1)
A191725=dispersion of A047208 (0, 4 mod 5 and >1)
A191726=dispersion of A047216 (1, 2 mod 5 and >1)
A191727=dispersion of A047219 (1, 3 mod 5 and >1)
A191728=dispersion of A047209 (1, 4 mod 5 and >1)
A191729=dispersion of A047221 (2, 3 mod 5 and >1)
A191730=dispersion of A047211 (2, 4 mod 5 and >1)
A191731=dispersion of A047204 (3, 4 mod 5 and >1)
...
A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
...
For further information about these 20 dispersions, see A191722.
...
Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

			Northwest corner:
1....2....5....12....30
3....7....17...42....105
4....10...25...62....155
6....15...37...92....230
8....20...50...125...312
9....22...55...137...342

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A047206, A047215, A191733, A191722, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=2; b=5; m[n_]:=If[Mod[n,2]==0,1,0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}]  (* A047215 *)
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}]] (* A191722 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191722  *)

A191724 Dispersion of A047218, (numbers >1 and congruent to 0 or 3 mod 5), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 8, 5, 4, 20, 13, 10, 6, 50, 33, 25, 15, 7, 125, 83, 63, 38, 18, 9, 313, 208, 158, 95, 45, 23, 11, 783, 520, 395, 238, 113, 58, 28, 12, 1958, 1300, 988, 595, 283, 145, 70, 30, 14, 4895, 3250, 2470, 1488, 708, 363, 175, 75, 35, 16, 12238, 8125, 6175

Offset: 1

Clark Kimberling, Jun 13 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
...
A191722=dispersion of A008851 (0, 1 mod 5 and >1)
A191723=dispersion of A047215 (0, 2 mod 5 and >1)
A191724=dispersion of A047218 (0, 3 mod 5 and >1)
A191725=dispersion of A047208 (0, 4 mod 5 and >1)
A191726=dispersion of A047216 (1, 2 mod 5 and >1)
A191727=dispersion of A047219 (1, 3 mod 5 and >1)
A191728=dispersion of A047209 (1, 4 mod 5 and >1)
A191729=dispersion of A047221 (2, 3 mod 5 and >1)
A191730=dispersion of A047211 (2, 4 mod 5 and >1)
A191731=dispersion of A047204 (3, 4 mod 5 and >1)
...
A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
...
For further information about these 20 dispersions, see A191722.
...
Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

			Northwest corner:
1....3....8....20....50
2....5....13...33....83
4....10...25...63....158
6....15...38...95....238
7....18...45...113...283
9....23...58...145...363

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A032793, A047218, A191734, A191722, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=3; b=5; m[n_]:=If[Mod[n,2]==0,1,0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}]  (* A047218 *)
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}]] (* A191724 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191724  *)

A191725 Dispersion of A047208, (numbers >1 and congruent to 0 or 4 mod 5), by antidiagonals.

Original entry on oeis.org

1, 4, 2, 10, 5, 3, 25, 14, 9, 6, 64, 35, 24, 15, 7, 160, 89, 60, 39, 19, 8, 400, 224, 150, 99, 49, 20, 11, 1000, 560, 375, 249, 124, 50, 29, 12, 2500, 1400, 939, 624, 310, 125, 74, 30, 13, 6250, 3500, 2349, 1560, 775, 314, 185, 75, 34, 16, 15625, 8750, 5874

Offset: 1

Clark Kimberling, Jun 13 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
...
A191722=dispersion of A008851 (0, 1 mod 5 and >1)
A191723=dispersion of A047215 (0, 2 mod 5 and >1)
A191724=dispersion of A047218 (0, 3 mod 5 and >1)
A191725=dispersion of A047208 (0, 4 mod 5 and >1)
A191726=dispersion of A047216 (1, 2 mod 5 and >1)
A191727=dispersion of A047219 (1, 3 mod 5 and >1)
A191728=dispersion of A047209 (1, 4 mod 5 and >1)
A191729=dispersion of A047221 (2, 3 mod 5 and >1)
A191730=dispersion of A047211 (2, 4 mod 5 and >1)
A191731=dispersion of A047204 (3, 4 mod 5 and >1)
...
A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
...
For further information about these 20 dispersions, see A191722.
...
Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

			Northwest corner:
1....4....10....25....64
2....5....14....35...89
3....9....24...60...150
6....15...39...99...249
7....19...49...124..310
8....20...50...125...314

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A047223, A047208, A191735, A191722, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=4; b=5; m[n_]:=If[Mod[n,2]==0,1,0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}]  (* A047208 *)
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}]] (* A191725 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191725  *)

A191726 Dispersion of A047216, (numbers >1 and congruent to 1 or 2 mod 5), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 6, 7, 4, 16, 17, 11, 5, 41, 42, 27, 12, 8, 102, 106, 67, 31, 21, 9, 256, 266, 167, 77, 52, 22, 10, 641, 666, 417, 192, 131, 56, 26, 13, 1602, 1666, 1042, 481, 327, 141, 66, 32, 14, 4006, 4166, 2606, 1202, 817, 352, 166, 81, 36, 15, 10016, 10416

Offset: 1

Clark Kimberling, Jun 13 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
...
A191722=dispersion of A008851 (0, 1 mod 5 and >1)
A191723=dispersion of A047215 (0, 2 mod 5 and >1)
A191724=dispersion of A047218 (0, 3 mod 5 and >1)
A191725=dispersion of A047208 (0, 4 mod 5 and >1)
A191726=dispersion of A047216 (1, 2 mod 5 and >1)
A191727=dispersion of A047219 (1, 3 mod 5 and >1)
A191728=dispersion of A047209 (1, 4 mod 5 and >1)
A191729=dispersion of A047221 (2, 3 mod 5 and >1)
A191730=dispersion of A047211 (2, 4 mod 5 and >1)
A191731=dispersion of A047204 (3, 4 mod 5 and >1)
...
A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
...
For further information about these 20 dispersions, see A191722.
...
Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

			Northwest corner:
1....2....6....16....41
3....7....17...42....106
4....11...27...67....167
5....12...31...77....192
8....21...52...131...327
9....22...56...141...352

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A047205, A047216, A191736, A191722, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=2; b=6; m[n_]:=If[Mod[n,2]==0,1,0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}]  (* A047216 *)
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}]] (* A191726 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191726  *)

A191727 Dispersion of A047219, (numbers >1 and congruent to 1 or 3 mod 5), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 8, 6, 4, 21, 16, 11, 5, 53, 41, 28, 13, 7, 133, 103, 71, 33, 18, 9, 333, 258, 178, 83, 46, 23, 10, 833, 646, 446, 208, 116, 58, 26, 12, 2083, 1616, 1116, 521, 291, 146, 66, 31, 14, 5208, 4041, 2791, 1303, 728, 366, 166, 78, 36, 15, 13021, 10103

Offset: 1

Clark Kimberling, Jun 13 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
...
A191722=dispersion of A008851 (0, 1 mod 5 and >1)
A191723=dispersion of A047215 (0, 2 mod 5 and >1)
A191724=dispersion of A047218 (0, 3 mod 5 and >1)
A191725=dispersion of A047208 (0, 4 mod 5 and >1)
A191726=dispersion of A047216 (1, 2 mod 5 and >1)
A191727=dispersion of A047219 (1, 3 mod 5 and >1)
A191728=dispersion of A047209 (1, 4 mod 5 and >1)
A191729=dispersion of A047221 (2, 3 mod 5 and >1)
A191730=dispersion of A047211 (2, 4 mod 5 and >1)
A191731=dispersion of A047204 (3, 4 mod 5 and >1)
...
A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
...
For further information about these 20 dispersions, see A191722.
...
Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

			Northwest corner:
1....3....8....21....53
2....6....16...41....103
4....11...28...71....178
5....13...33...83....208
7....18...46...116...291
9....23...58...146...366

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

Cf. A047212, A047219, A191737, A191722, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=3; b=6; m[n_]:=If[Mod[n,2]==0,1,0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}]  (* A047219 *)
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}]] (* A191727 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191727  *)

A191728 Dispersion of A047209, (numbers >1 and congruent to 1 or 4 mod 5), by antidiagonals.

Original entry on oeis.org

1, 4, 2, 11, 6, 3, 29, 16, 9, 5, 74, 41, 24, 14, 7, 186, 104, 61, 36, 19, 8, 466, 261, 154, 91, 49, 21, 10, 1166, 654, 386, 229, 124, 54, 26, 12, 2916, 1636, 966, 574, 311, 136, 66, 31, 13, 7291, 4091, 2416, 1436, 779, 341, 166, 79, 34, 15, 18229, 10229

Offset: 1

Clark Kimberling, Jun 13 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
...
A191722=dispersion of A008851 (0, 1 mod 5 and >1)
A191723=dispersion of A047215 (0, 2 mod 5 and >1)
A191724=dispersion of A047218 (0, 3 mod 5 and >1)
A191725=dispersion of A047208 (0, 4 mod 5 and >1)
A191726=dispersion of A047216 (1, 2 mod 5 and >1)
A191727=dispersion of A047219 (1, 3 mod 5 and >1)
A191728=dispersion of A047209 (1, 4 mod 5 and >1)
A191729=dispersion of A047221 (2, 3 mod 5 and >1)
A191730=dispersion of A047211 (2, 4 mod 5 and >1)
A191731=dispersion of A047204 (3, 4 mod 5 and >1)
...
A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
...
For further information about these 20 dispersions, see A191722.
...
Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

			Northwest corner:
1....4....11...29....74
2....6....16...41....104
3....9....24...61....154
5....14...36...91....229
7....19...49...124...311
8....21...54...136...341

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A047222, A047209, A191738, A191722, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=4; b=6; m[n_]:=If[Mod[n,2]==0,1,0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}]  (* A047209 *)
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}]] (* A191728 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191728  *)

A191729 Dispersion of A047221, (numbers >1 and congruent to 2 or 3 mod 5), by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 8, 5, 7, 18, 12, 6, 17, 43, 28, 13, 9, 42, 107, 68, 32, 22, 10, 103, 267, 168, 78, 53, 23, 11, 257, 667, 418, 193, 132, 57, 27, 14, 642, 1667, 1043, 482, 328, 142, 67, 33, 15, 1603, 4167, 2607, 1203, 818, 353, 167, 82, 37, 16, 4007, 10417, 6517

Offset: 1

Clark Kimberling, Jun 13 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
...
A191722=dispersion of A008851 (0, 1 mod 5 and >1)
A191723=dispersion of A047215 (0, 2 mod 5 and >1)
A191724=dispersion of A047218 (0, 3 mod 5 and >1)
A191725=dispersion of A047208 (0, 4 mod 5 and >1)
A191726=dispersion of A047216 (1, 2 mod 5 and >1)
A191727=dispersion of A047219 (1, 3 mod 5 and >1)
A191728=dispersion of A047209 (1, 4 mod 5 and >1)
A191729=dispersion of A047221 (2, 3 mod 5 and >1)
A191730=dispersion of A047211 (2, 4 mod 5 and >1)
A191731=dispersion of A047204 (3, 4 mod 5 and >1)
...
A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
...
For further information about these 20 dispersions, see A191722.
...
Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

			Northwest corner:
1....2....3....7.....17
4....8....18...43....107
5....12...28...68....168
6....13...32...78....193
9....22...53...132...328
10...23...57...142...353

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A008854, A047221, A191739, A191722, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=2; b=3; m[n_]:=If[Mod[n,2]==0,1,0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}]  (* A047221 *)
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}]] (* A191729 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191729  *)

A191730 Dispersion of A047211, (numbers >1 and congruent to 2 or 4 mod 5), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 9, 17, 12, 6, 22, 42, 29, 14, 8, 54, 104, 72, 34, 19, 10, 134, 259, 179, 84, 47, 24, 11, 334, 647, 447, 209, 117, 59, 27, 13, 834, 1617, 1117, 522, 292, 147, 67, 32, 15, 2084, 4042, 2792, 1304, 729, 367, 167, 79, 37, 16, 5209, 10104, 6979

Offset: 1

Clark Kimberling, Jun 13 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
...
A191722=dispersion of A008851 (0, 1 mod 5 and >1)
A191723=dispersion of A047215 (0, 2 mod 5 and >1)
A191724=dispersion of A047218 (0, 3 mod 5 and >1)
A191725=dispersion of A047208 (0, 4 mod 5 and >1)
A191726=dispersion of A047216 (1, 2 mod 5 and >1)
A191727=dispersion of A047219 (1, 3 mod 5 and >1)
A191728=dispersion of A047209 (1, 4 mod 5 and >1)
A191729=dispersion of A047221 (2, 3 mod 5 and >1)
A191730=dispersion of A047211 (2, 4 mod 5 and >1)
A191731=dispersion of A047204 (3, 4 mod 5 and >1)
...
A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
...
For further information about these 20 dispersions, see A191722.
...
Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

			Northwest corner:
1....2....4.....9.....22
3....7....17...42....104
5....12...29...72....179
6....14...34...84....209
8....19...47...117...292
10...24...59...147...367

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

Cf. A047220, A047211, A191740, A191722, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=2; b=4; m[n_]:=If[Mod[n,2]==0,1,0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}]  (* A047211 *)
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}]] (* A191730 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191730  *)

cp's OEIS Frontend

A191702 Dispersion of A008587 (5,10,15,20,25,30,...), by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A191722 Dispersion of A008851, (numbers >1 and congruent to 0 or 1 mod 5), by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A191723 Dispersion of A047215, (numbers >1 and congruent to 0 or 2 mod 5), by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A191724 Dispersion of A047218, (numbers >1 and congruent to 0 or 3 mod 5), by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A191725 Dispersion of A047208, (numbers >1 and congruent to 0 or 4 mod 5), by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A191726 Dispersion of A047216, (numbers >1 and congruent to 1 or 2 mod 5), by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A191727 Dispersion of A047219, (numbers >1 and congruent to 1 or 3 mod 5), by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A191728 Dispersion of A047209, (numbers >1 and congruent to 1 or 4 mod 5), by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs