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A191673
Dispersion of A004773 (>1 and congruent to 0 or 1 or 2 mod 4), by antidiagonals.
17
1, 2, 3, 4, 5, 7, 6, 8, 10, 11, 9, 12, 14, 16, 15, 13, 17, 20, 22, 21, 19, 18, 24, 28, 30, 29, 26, 23, 25, 33, 38, 41, 40, 36, 32, 27, 34, 45, 52, 56, 54, 49, 44, 37, 31, 46, 61, 70, 76, 73, 66, 60, 50, 42, 35, 62, 82, 94, 102, 98, 89, 81, 68, 57, 48, 39, 84
OFFSET
1,2
COMMENTS
For a background discussion of dispersions, see A191426.
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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:
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A191452=dispersion of A008586 (4k, k>=1)
A191667=dispersion of A016813 (4k+1, k>=1)
A191668=dispersion of A016825 (4k+2, k>=0)
A191669=dispersion of A004767 (4k+3, k>=0)
A191670=dispersion of A042968 (1 or 2 or 3 mod 4 and >=2)
A191671=dispersion of A004772 (0 or 1 or 3 mod 4 and >=2)
A191672=dispersion of A004773 (0 or 1 or 2 mod 4 and >=2)
A191673=dispersion of A004773 (0 or 2 or 3 mod 4 and >=2)
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EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191452 has 1st col A042968, all else A008486
A191667 has 1st col A004772, all else A016813
A191668 has 1st col A042965, all else A016825
A191669 has 1st col A004773, all else A004767
A191670 has 1st col A008486, all else A042968
A191671 has 1st col A016813, all else A004772
A191672 has 1st col A016825, all else A042965
A191673 has 1st col A004767, all else A004773
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Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c mod m)", (as in the Mathematica program below):
If f(n)=(n mod 3), then (a,b,c,a,b,c,a,b,c,...) is given by
a*f(n+2)+b*f(n+1)+c*f(n), so that "(a or b or c mod m)" is given by
a*f(n+2)+b*f(n+1)+c*f(n)+m*floor((n-1)/3)), for n>=1.
LINKS
Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)
EXAMPLE
Northwest corner:
1....2....4....6....9
3....5....8....12...17
7....10...14...20...28
11...16...22...30...41
15...21...29...40...54
MATHEMATICA
(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
a = 2; b = 4; c2 = 5; m[n_] := If[Mod[n, 3] == 0, 1, 0];
f[n_] := a*m[n + 2] + b*m[n + 1] + c2*m[n] + 4*Floor[(n - 1)/3]
Table[f[n], {n, 1, 30}] (* A004773 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191673 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191673 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jun 11 2011
STATUS
approved