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A191740
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Dispersion of A047220, (numbers >1 and congruent to 0 or 1 or 3 mod 5), by antidiagonals.
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20
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1, 3, 2, 6, 5, 4, 11, 10, 8, 7, 20, 18, 15, 13, 9, 35, 31, 26, 23, 16, 12, 60, 53, 45, 40, 28, 21, 14, 101, 90, 76, 68, 48, 36, 25, 17, 170, 151, 128, 115, 81, 61, 43, 30, 19, 285, 253, 215, 193, 136, 103, 73, 51, 33, 22, 476, 423, 360, 323, 228, 173, 123
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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....6....11...20
2....5....10...18...31
4....8....15...26...45
7....13...23...40...68
9....16...28...48...81
12...21...36...61...103
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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; c2=6; 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}] (* A047220 *)
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}]] (* A191740 *)
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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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