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A191663
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Dispersion of A042948 (numbers >3, congruent to 0 or 1 mod 4), by antidiagonals.
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34
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1, 4, 2, 9, 5, 3, 20, 12, 8, 6, 41, 25, 17, 13, 7, 84, 52, 36, 28, 16, 10, 169, 105, 73, 57, 33, 21, 11, 340, 212, 148, 116, 68, 44, 24, 14, 681, 425, 297, 233, 137, 89, 49, 29, 15, 1364, 852, 596, 468, 276, 180, 100, 60, 32, 18, 2729, 1705, 1193, 937, 553
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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, see A191426.
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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:
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EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
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There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.
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LINKS
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EXAMPLE
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Northwest corner:
1...4...9....20...41
2...5...12...25...52
3...8...17...36...73
6...13..28...57...116
7...16..33...68...137
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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] + 4*Floor[(n - 1)/2]
Table[f[n], {n, 1, 30}] (* A042948: (4+4k, 5+4k) *)
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}]] (* A191663 *)
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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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