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A191665 Dispersion of A042963 (numbers >1, congruent to 1 or 2 mod 4), by antidiagonals.

%I A191665 #7 Oct 18 2017 17:18:55
%S A191665 1,2,3,5,6,4,10,13,9,7,21,26,18,14,8,42,53,37,29,17,11,85,106,74,58,
%T A191665 34,22,12,170,213,149,117,69,45,25,15,341,426,298,234,138,90,50,30,16,
%U A191665 682,853,597,469,277,181,101,61,33,19,1365,1706,1194,938,554
%N A191665 Dispersion of A042963 (numbers >1, congruent to 1 or 2 mod 4), by antidiagonals.
%C A191665 Row 1:  A000975
%C A191665 Row 2:  A081254
%C A191665 Row 3:  A081253
%C A191665 Row 4:  A052997
%C A191665 For a background discussion of dispersions, see A191426.
%C A191665 ...
%C A191665 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 A191665 ...
%C A191665 A191663=dispersion of A042948 (0 or 1 mod 4 and >1)
%C A191665 A054582=dispersion of A005843 (0 or 2 mod 4 and >1; evens)
%C A191665 A191664=dispersion of A014601 (0 or 3 mod 4 and >1)
%C A191665 A191665=dispersion of A042963 (1 or 2 mod 4 and >1)
%C A191665 A191448=dispersion of A005408 (1 or 3 mod 4 and >1, odds)
%C A191665 A191666=dispersion of A042964 (2 or 3 mod 4)
%C A191665 ...
%C A191665 EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
%C A191665 A191663 has 1st col A042964, all else A042948
%C A191665 A054582 has 1st col A005408, all else A005843
%C A191665 A191664 has 1st col A042963, all else A014601
%C A191665 A191665 has 1st col A014601, all else A042963
%C A191665 A191448 has 1st col A005843, all else A005408
%C A191665 A191666 has 1st col A042948, all else A042964
%C A191665 ...
%C A191665 There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
%C A191665    If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
%C A191665    a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
%C A191665    a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.
%H A191665 Ivan Neretin, <a href="/A191665/b191665.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e A191665 Northwest corner:
%e A191665 1...2...5....10...21
%e A191665 3...6...13...26...53
%e A191665 4...9...18...37...74
%e A191665 7...14..29...58...117
%e A191665 8...17..34...69...138
%t A191665 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191665 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191665 a = 2; b = 5; m[n_] := If[Mod[n, 2] == 0, 1, 0];
%t A191665 f[n_] := a*m[n + 1] + b*m[n] + 4*Floor[(n - 1)/2]
%t A191665 Table[f[n], {n, 1, 30}]  (* A042963: (2+4k,5+4k) *)
%t A191665 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191665 rows = {NestList[f, 1, c]};
%t A191665 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191665 t[i_, j_] := rows[[i, j]];
%t A191665 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191665 (* A191665 *)
%t A191665 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
%t A191665 (* A191665  *)
%Y A191665 Cf. A042963, A014601, A191426, A191664.
%K A191665 nonn,tabl
%O A191665 1,2
%A A191665 _Clark Kimberling_, Jun 11 2011