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A191668
Dispersion of A016825 (4k+2, k>0), by antidiagonals.
8
1, 2, 3, 6, 10, 4, 22, 38, 14, 5, 86, 150, 54, 18, 7, 342, 598, 214, 70, 26, 8, 1366, 2390, 854, 278, 102, 30, 9, 5462, 9558, 3414, 1110, 406, 118, 34, 11, 21846, 38230, 13654, 4438, 1622, 470, 134, 42, 12, 87382, 152918, 54614, 17750, 6486, 1878, 534, 166
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)
FORMULA
Conjecture: a(n,k) = (8 + (3*floor((4*n + 1)/3) - 2)*4^k)/12 = (8 + (3*A042965(n+1) - 2)*A000302(k))/12. - L. Edson Jeffery, Feb 14 2015
EXAMPLE
Northwest corner:
. 1 2 6 22 86 342 1366 5462 21846 87382
. 3 10 38 150 598 2390 9558 38230 152918 611670
. 4 14 54 214 854 3414 13654 54614 218454 873814
. 5 18 70 278 1110 4438 17750 70998 283990 1135958
. 7 26 102 406 1622 6486 25942 103766 415062 1660246
. 8 30 118 470 1878 7510 30038 120150 480598 1922390
. 9 34 134 534 2134 8534 34134 136534 546134 2184534
. 11 42 166 662 2646 10582 42326 169302 677206 2708822
. 12 46 182 726 2902 11606 46422 185686 742742 2970966
. 13 50 198 790 3158 12630 50518 202070 808278 3233110
MATHEMATICA
(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
f[n_] := 4*n-2
Table[f[n], {n, 1, 30}] (* A016825 *)
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}]] (* A191668 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191668 *)
(* Conjectured: *) Grid[Table[(8 + (3*Floor[(4*n + 1)/3] - 2)*4^k)/12, {n, 10}, {k, 10}]] (* L. Edson Jeffery, Feb 14 2015 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jun 11 2011
STATUS
approved