A191665 - OEIS

%I #7 Oct 18 2017 17:18:55

%S 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 34,22,12,170,213,149,117,69,45,25,15,341,426,298,234,138,90,50,30,16,

%U 682,853,597,469,277,181,101,61,33,19,1365,1706,1194,938,554

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

%C Row 1:  A000975

%C Row 2:  A081254

%C Row 3:  A081253

%C Row 4:  A052997

%C For a background discussion of dispersions, see A191426.

%C ...

%C 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 ...

%C A191663=dispersion of A042948 (0 or 1 mod 4 and >1)

%C A054582=dispersion of A005843 (0 or 2 mod 4 and >1; evens)

%C A191664=dispersion of A014601 (0 or 3 mod 4 and >1)

%C A191665=dispersion of A042963 (1 or 2 mod 4 and >1)

%C A191448=dispersion of A005408 (1 or 3 mod 4 and >1, odds)

%C A191666=dispersion of A042964 (2 or 3 mod 4)

%C ...

%C EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):

%C A191663 has 1st col A042964, all else A042948

%C A054582 has 1st col A005408, all else A005843

%C A191664 has 1st col A042963, all else A014601

%C A191665 has 1st col A014601, all else A042963

%C A191448 has 1st col A005843, all else A005408

%C A191666 has 1st col A042948, all else A042964

%C ...

%C There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):

%C    If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by

%C    a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by

%C    a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.

%H Ivan Neretin, <a href="/A191665/b191665.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals, flattened)

%e Northwest corner:

%e 1...2...5....10...21

%e 3...6...13...26...53

%e 4...9...18...37...74

%e 7...14..29...58...117

%e 8...17..34...69...138

%t (* Program generates the dispersion array T of the increasing sequence f[n] *)

%t r = 40; r1 = 12;  c = 40; c1 = 12;

%t a = 2; b = 5; m[n_] := If[Mod[n, 2] == 0, 1, 0];

%t f[n_] := a*m[n + 1] + b*m[n] + 4*Floor[(n - 1)/2]

%t Table[f[n], {n, 1, 30}]  (* A042963: (2+4k,5+4k) *)

%t mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]

%t rows = {NestList[f, 1, c]};

%t Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];

%t t[i_, j_] := rows[[i, j]];

%t TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]

%t (* A191665 *)

%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]

%t (* A191665  *)

%Y Cf. A042963, A014601, A191426, A191664.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Jun 11 2011