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A191666 Dispersion of A042964 (numbers congruent to 2 or 3 mod 4), by antidiagonals.

%I A191666 #7 Oct 18 2017 17:19:14
%S A191666 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,
%T A191666 35,23,13,86,214,150,118,70,46,26,16,171,427,299,235,139,91,51,31,17,
%U A191666 342,854,598,470,278,182,102,62,34,20,683,1707,1195,939,555,363
%N A191666 Dispersion of A042964 (numbers congruent to 2 or 3 mod 4), by antidiagonals.
%C A191666 Row 1:  A005578
%C A191666 Row 2:  A160113
%C A191666 For a background discussion of dispersions, see A191426.
%C A191666 ...
%C A191666 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:
%C A191666 ...
%C A191666 A191663=dispersion of A042948 (0 or 1 mod 4 and >1)
%C A191666 A054582=dispersion of A005843 (0 or 2 mod 4 and >1; evens)
%C A191666 A191664=dispersion of A014601 (0 or 3 mod 4 and >1)
%C A191666 A191665=dispersion of A042963 (1 or 2 mod 4 and >1)
%C A191666 A191448=dispersion of A005408 (1 or 3 mod 4 and >1, odds)
%C A191666 A191666=dispersion of A042964 (2 or 3 mod 4)
%C A191666 ...
%C A191666 EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
%C A191666 A191663 has 1st col A042964, all else A042948
%C A191666 A054582 has 1st col A005408, all else A005843
%C A191666 A191664 has 1st col A042963, all else A014601
%C A191666 A191665 has 1st col A014601, all else A042963
%C A191666 A191448 has 1st col A005843, all else A005408
%C A191666 A191666 has 1st col A042948, all else A042964
%C A191666 ...
%C A191666 There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
%C A191666    If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
%C A191666    a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
%C A191666    a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.
%H A191666 Ivan Neretin, <a href="/A191666/b191666.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e A191666 Northwest corner:
%e A191666 1...2...3....6...11
%e A191666 4...7...14....27...54
%e A191666 5...10...19...38...75
%e A191666 8...15..30...59...118
%e A191666 8...18..35...70...139
%t A191666 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191666 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191666 a = 2; b = 3; m[n_] := If[Mod[n, 2] == 0, 1, 0];
%t A191666 f[n_] := a*m[n + 1] + b*m[n] + 4*Floor[(n - 1)/2]
%t A191666 Table[f[n], {n, 1, 30}]  (* A042964: (2+4k,3+4k) *)
%t A191666 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191666 rows = {NestList[f, 1, c]};
%t A191666 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191666 t[i_, j_] := rows[[i, j]];
%t A191666 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191666 *)
%t A191666 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191666  *)
%Y A191666 Cf. A042963, A014601, A191426, A191663.
%K A191666 nonn,tabl
%O A191666 1,2
%A A191666 _Clark Kimberling_, Jun 11 2011