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A191663 Dispersion of A042948 (numbers >3, congruent to 0 or 1 mod 4), by antidiagonals.

%I A191663 #10 Oct 18 2017 18:30:27
%S A191663 1,4,2,9,5,3,20,12,8,6,41,25,17,13,7,84,52,36,28,16,10,169,105,73,57,
%T A191663 33,21,11,340,212,148,116,68,44,24,14,681,425,297,233,137,89,49,29,15,
%U A191663 1364,852,596,468,276,180,100,60,32,18,2729,1705,1193,937,553
%N A191663 Dispersion of A042948 (numbers >3, congruent to 0 or 1 mod 4), by antidiagonals.
%C A191663 Row 1: A084639.
%C A191663 For a background discussion of dispersions, see A191426.
%C A191663 ...
%C A191663 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 A191663 ...
%C A191663 A191663=dispersion of A042948 (0 or 1 mod 4 and >1)
%C A191663 A054582=dispersion of A005843 (0 or 2 mod 4 and >1; evens)
%C A191663 A191664=dispersion of A014601 (0 or 3 mod 4 and >1)
%C A191663 A191665=dispersion of A042963 (1 or 2 mod 4 and >1)
%C A191663 A191448=dispersion of A005408 (1 or 3 mod 4 and >1, odds)
%C A191663 A191666=dispersion of A042964 (2 or 3 mod 4)
%C A191663 ...
%C A191663 EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
%C A191663 A191663 has 1st col A042964, all else A042948
%C A191663 A054582 has 1st col A005408, all else A005843
%C A191663 A191664 has 1st col A042963, all else A014601
%C A191663 A191665 has 1st col A014601, all else A042963
%C A191663 A191448 has 1st col A005843, all else A005408
%C A191663 A191666 has 1st col A042948, all else A042964
%C A191663 ...
%C A191663 There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
%C A191663    If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
%C A191663    a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
%C A191663    a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.
%H A191663 Ivan Neretin, <a href="/A191663/b191663.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e A191663 Northwest corner:
%e A191663 1...4...9....20...41
%e A191663 2...5...12...25...52
%e A191663 3...8...17...36...73
%e A191663 6...13..28...57...116
%e A191663 7...16..33...68...137
%t A191663 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191663 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191663 a = 4; b = 5; m[n_] := If[Mod[n, 2] == 0, 1, 0];
%t A191663 f[n_] := a*m[n + 1] + b*m[n] + 4*Floor[(n - 1)/2]
%t A191663 Table[f[n], {n, 1, 30}]  (* A042948: (4+4k,5+4k) *)
%t A191663 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191663 rows = {NestList[f, 1, c]};
%t A191663 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191663 t[i_, j_] := rows[[i, j]];
%t A191663 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191663 (* A191663 *)
%t A191663 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191663 *)
%Y A191663 Cf. A042948, A042964, A191426.
%K A191663 nonn,tabl
%O A191663 1,2
%A A191663 _Clark Kimberling_, Jun 11 2011