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A191711
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Dispersion of A001068, (numbers >1 and congruent to 0, 1, 2, or 3 mod 5), by antidiagonals.
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10
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1, 2, 4, 3, 6, 9, 5, 8, 12, 14, 7, 11, 16, 18, 19, 10, 15, 21, 23, 25, 24, 13, 20, 27, 30, 32, 31, 29, 17, 26, 35, 38, 41, 40, 37, 34, 22, 33, 45, 48, 52, 51, 47, 43, 39, 28, 42, 57, 61, 66, 65, 60, 55, 50, 44, 36, 53, 72, 77, 83, 82, 76, 70, 63, 56, 49, 46
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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 or mod 4, see A191655, A191663, A191667.
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Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten 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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Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
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If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).
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LINKS
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EXAMPLE
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Northwest corner:
1....2....3....5....7
4....6....8....11...15
9....12...16...21...27
14...18...23...30...38
19...25...32...41...52
24...31...40...51...65
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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; c2=5; d=6; m[n_]:=If[Mod[n, 4]==0, 1, 0];
f[n_]:=a*m[n+3]+b*m[n+2]+c2*m[n+1]+d*m[n]+5*Floor[(n-1)/4]
Table[f[n], {n, 1, 30}] (* A001068 *)
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}]] (* A191711 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191711 *)
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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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