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A191666
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Dispersion of A042964 (numbers congruent to 2 or 3 mod 4), by antidiagonals.
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4
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1, 2, 4, 3, 7, 5, 6, 14, 10, 8, 11, 27, 19, 15, 9, 22, 54, 38, 30, 18, 12, 43, 107, 75, 59, 35, 23, 13, 86, 214, 150, 118, 70, 46, 26, 16, 171, 427, 299, 235, 139, 91, 51, 31, 17, 342, 854, 598, 470, 278, 182, 102, 62, 34, 20, 683, 1707, 1195, 939, 555, 363
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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...2...3....6...11
4...7...14....27...54
5...10...19...38...75
8...15..30...59...118
8...18..35...70...139
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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] + 4*Floor[(n - 1)/2]
Table[f[n], {n, 1, 30}] (* A042964: (2+4k, 3+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}]] (* A191666 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191666 *)
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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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