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A191656 Dispersion of (2,4,5,7,8,10,...), by antidiagonals.

%I A191656 #6 Mar 30 2012 18:57:32
%S A191656 1,2,3,4,5,6,7,8,10,9,11,13,16,14,12,17,20,25,22,19,15,26,31,38,34,29,
%T A191656 23,18,40,47,58,52,44,35,28,21,61,71,88,79,67,53,43,32,24,92,107,133,
%U A191656 119,101,80,65,49,37,27,139,161,200,179,152,121,98,74,56
%N A191656 Dispersion of (2,4,5,7,8,10,...), by antidiagonals.
%C A191656 Row 1:  A006999.
%C A191656 For a background discussion of dispersions, see A191426.
%C A191656 ...
%C A191656 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 A191656 ...
%C A191656 A191449=dispersion of A008583 (0 mod 3)
%C A191656 A191451=dispersion of A016777 (1 mod 3)
%C A191656 A191450=dispersion of A016789 (2 mod 3)
%C A191656 A191656=dispersion of A001651 (1 or 2 mod 3)
%C A191656 A083044=dispersion of A007494 (0 or 2 mod 3)
%C A191656 A191655=dispersion of A032766 (0 or 1 mod 3)
%C A191656 ...
%C A191656 EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
%C A191656 A191449 has 1st col A001651, all else A008583
%C A191656 A191451 has 1st col A007494, all else A016777
%C A191656 A191450 has 1st col A032766, all else A016789
%C A191656 A191656 has 1st col A008583, all else A001651
%C A191656 A083044 has 1st col A016777, all else A083044
%C A191656 A191655 has 1st col A016789, all else A032766
%C A191656 ...
%C A191656 There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
%C A191656    If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
%C A191656    a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
%C A191656    a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.
%e A191656 Northwest corner:
%e A191656 1...2....4....7....11
%e A191656 3...5....8....13...20
%e A191656 6...10...16...25...38
%e A191656 9...14...22...34...52
%e A191656 12..19...29...44...67
%t A191656 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191656 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191656 a = 2; b = 4; m[n_] := If[Mod[n, 2] == 0, 1, 0];
%t A191656 f[n_] := a*m[n + 1] + b*m[n] + 3*Floor[(n - 1)/2]
%t A191656 Table[f[n], {n, 1, 30}]  (* A001651: (2+5k,4+5k, k>=0) *)
%t A191656 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191656 rows = {NestList[f, 1, c]};
%t A191656 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191656 t[i_, j_] := rows[[i, j]];
%t A191656 TableForm[Table[t[i, j], {i,1,10}, {j,1,10}]]          (* A191656 array *)
%t A191656 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]   (* A191656 sequence *)
%Y A191656 Cf. A001651, A008583, A191426.
%K A191656 nonn,tabl
%O A191656 1,2
%A A191656 _Clark Kimberling_, Jun 10 2011