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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
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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
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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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