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A191664 Dispersion of A014601 (numbers >2, congruent to 0 or 3 mod 4), by antidiagonals.

%I A191664 #20 Oct 18 2017 17:19:32
%S A191664 1,3,2,7,4,5,15,8,11,6,31,16,23,12,9,63,32,47,24,19,10,127,64,95,48,
%T A191664 39,20,13,255,128,191,96,79,40,27,14,511,256,383,192,159,80,55,28,17,
%U A191664 1023,512,767,384,319,160,111,56,35,18,2047,1024,1535,768,639
%N A191664 Dispersion of A014601 (numbers >2, congruent to 0 or 3 mod 4), by antidiagonals.
%C A191664 Row 1:  A000225 (-1+2^n)
%C A191664 Row 2:  A000079 (2^n)
%C A191664 Row 3:  A055010
%C A191664 Row 4:  3*A000079
%C A191664 Row 5:  A153894
%C A191664 Row 6:  5*A000079
%C A191664 Row 7:  A086224
%C A191664 Row 8:  A005009
%C A191664 Row 9:  A052996
%C A191664 For a background discussion of dispersions, see A191426.
%C A191664 ...
%C A191664 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 A191664 ...
%C A191664 A191663=dispersion of A042948 (0 or 1 mod 4 and >1)
%C A191664 A054582=dispersion of A005843 (0 or 2 mod 4 and >1; evens)
%C A191664 A191664=dispersion of A014601 (0 or 3 mod 4 and >1)
%C A191664 A191665=dispersion of A042963 (1 or 2 mod 4 and >1)
%C A191664 A191448=dispersion of A005408 (1 or 3 mod 4 and >1, odds)
%C A191664 A191666=dispersion of A042964 (2 or 3 mod 4)
%C A191664 ...
%C A191664 EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
%C A191664 A191663 has 1st col A042964, all else A042948
%C A191664 A054582 has 1st col A005408, all else A005843
%C A191664 A191664 has 1st col A042963, all else A014601
%C A191664 A191665 has 1st col A014601, all else A042963
%C A191664 A191448 has 1st col A005843, all else A005408
%C A191664 A191666 has 1st col A042948, all else A042964
%C A191664 ...
%C A191664 There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
%C A191664    If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
%C A191664    a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
%C A191664    a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.
%C A191664 This sequence is a permutation of the natural numbers.  - _L. Edson Jeffery_, Aug 13 2014
%H A191664 Ivan Neretin, <a href="/A191664/b191664.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)
%e A191664 Northwest corner:
%e A191664 1...3...7....15...31
%e A191664 2...4...8....16...32
%e A191664 5...11..23...47...95
%e A191664 6...12..24...48...96
%e A191664 9...19..39...79...159
%t A191664 (* Program generates the dispersion array T of the increasing sequence f[n] *)
%t A191664 r = 40; r1 = 12;  c = 40; c1 = 12;
%t A191664 a = 3; b = 4; m[n_] := If[Mod[n, 2] == 0, 1, 0];
%t A191664 f[n_] := a*m[n + 1] + b*m[n] + 4*Floor[(n - 1)/2]
%t A191664 Table[f[n], {n, 1, 30}]  (* A014601(n+2): (4+4k,5+4k) *)
%t A191664 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191664 rows = {NestList[f, 1, c]};
%t A191664 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191664 t[i_, j_] := rows[[i, j]];
%t A191664 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191664 *)
%t A191664 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191664  *)
%t A191664 (* _Clark Kimberling_, Jun 11 2011 *)
%t A191664 Grid[Table[2^k*(2*Floor[(n + 1)/2] - 1) - Mod[n, 2], {n, 12}, {k, 12}]] (* _L. Edson Jeffery_, Aug 13 2014 *)
%Y A191664 Cf. A042963, A014601, A191665, A191426.
%Y A191664 Cf. A241957.
%K A191664 nonn,tabl
%O A191664 1,2
%A A191664 _Clark Kimberling_, Jun 11 2011