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%I #6 Mar 30 2012 18:57:32
%S 1,3,2,6,4,5,10,7,9,8,16,12,15,13,11,25,19,24,21,18,14,39,30,37,33,28,
%T 22,17,60,46,57,51,43,34,27,20,91,70,87,78,66,52,42,31,23,138,106,132,
%U 118,100,79,64,48,36,26,208,160,199,178,151,120,97,73,55
%N Dispersion of (2,5,8,11,14,17,...), by antidiagonals.
%C Row 1: A152009.
%C For a background discussion of dispersions, see A191426.
%C ...
%C Each of the sequences (3n, n>0), (3n+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:
%C ...
%C A191449=dispersion of A008583 (0 mod 3)
%C A191451=dispersion of A016777 (1 mod 3)
%C A191450=dispersion of A016789 (2 mod 3)
%C A191656=dispersion of A001651 (1 or 2 mod 3)
%C A083044=dispersion of A007494 (0 or 2 mod 3)
%C A191655=dispersion of A032766 (0 or 1 mod 3)
%C ...
%C EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
%C A191449 has 1st col A001651, all else A008583
%C A191451 has 1st col A007494, all else A016777
%C A191450 has 1st col A032766, all else A016789
%C A191656 has 1st col A008583, all else A001651
%C A083044 has 1st col A016777, all else A083044
%C A191655 has 1st col A016789, all else A032766
%C ...
%C There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
%C If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
%C a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
%C a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.
%e Northwest corner:
%e 1...3...6....10...16
%e 2...4...7....12...19
%e 5...9...15...24...37
%e 8...13..21...33...51
%e 11..18..28...43...66
%t (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t r = 40; r1 = 12; c = 40; c1 = 12;
%t a = 3; b = 4; m[n_] := If[Mod[n, 2] == 0, 1, 0];
%t f[n_] := a*m[n + 1] + b*m[n] + 3*Floor[(n - 1)/2]
%t Table[f[n], {n, 1, 30}] (* A032766: (3+5k,4+5k, k>=0) *)
%t mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t rows = {NestList[f, 1, c]};
%t Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t t[i_, j_] := rows[[i, j]];
%t TableForm[Table[t[i, j], {i,1,10}, {j,1,10}]] (* A191655 array *)
%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191655 sequence *)
%Y Cf. A016789, A032766, A191426.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Jun 10 2011
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