Search: a047205 -id:a047205
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A191736
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Dispersion of A047205, (numbers >1 and congruent to 0 or 3 or 4 mod 5), by antidiagonals.
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+20
20
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1, 3, 2, 5, 4, 6, 9, 8, 10, 7, 15, 14, 18, 13, 11, 25, 24, 30, 23, 19, 12, 43, 40, 50, 39, 33, 20, 16, 73, 68, 84, 65, 55, 34, 28, 17, 123, 114, 140, 109, 93, 58, 48, 29, 21, 205, 190, 234, 183, 155, 98, 80, 49, 35, 22, 343, 318, 390, 305, 259, 164, 134, 83
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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.
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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:
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For further information about these 20 dispersions, see A191722.
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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.
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LINKS
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EXAMPLE
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Northwest corner:
1....3....5....9....15
2....4....8....14...24
6....10...18...30...50
7....13...23...39...65
11...19...33...55...93
12...20...34...58...98
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MATHEMATICA
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(* 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]+5*Floor[(n-1)/3]
Table[f[n], {n, 1, 30}] (* A047205 *)
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}]] (* A191736 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191736 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A191722
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Dispersion of A008851, (numbers >1 and congruent to 0 or 1 mod 5), by antidiagonals.
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+10
20
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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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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.
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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:
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EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
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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.
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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(* 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}]]
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191722 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A191723
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Dispersion of A047215, (numbers >1 and congruent to 0 or 2 mod 5), by antidiagonals.
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+10
20
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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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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.
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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:
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For further information about these 20 dispersions, see A191722.
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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.
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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(* 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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A191724
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Dispersion of A047218, (numbers >1 and congruent to 0 or 3 mod 5), by antidiagonals.
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+10
20
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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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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.
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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:
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For further information about these 20 dispersions, see A191722.
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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.
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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(* 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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A191725
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Dispersion of A047208, (numbers >1 and congruent to 0 or 4 mod 5), by antidiagonals.
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+10
20
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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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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:
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For further information about these 20 dispersions, see A191722.
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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.
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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(* 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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A191726
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Dispersion of A047216, (numbers >1 and congruent to 1 or 2 mod 5), by antidiagonals.
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+10
20
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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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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:
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For further information about these 20 dispersions, see A191722.
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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.
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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(* 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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A191727
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Dispersion of A047219, (numbers >1 and congruent to 1 or 3 mod 5), by antidiagonals.
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+10
20
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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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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:
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...
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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.
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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(* 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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A191728
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Dispersion of A047209, (numbers >1 and congruent to 1 or 4 mod 5), by antidiagonals.
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+10
20
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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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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:
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...
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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.
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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(* 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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A191729
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Dispersion of A047221, (numbers >1 and congruent to 2 or 3 mod 5), by antidiagonals.
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+10
20
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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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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.
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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:
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For further information about these 20 dispersions, see A191722.
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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.
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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(* 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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A191730
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Dispersion of A047211, (numbers >1 and congruent to 2 or 4 mod 5), by antidiagonals.
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+10
20
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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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
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:
...
...
...
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.
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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(* 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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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