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A191732 Dispersion of A047202, (numbers >1 and congruent to 2 or 3 or 4 mod 5), by antidiagonals. 20

%I #11 Oct 11 2017 18:13:33

%S 1,2,5,3,8,6,4,13,9,10,7,22,14,17,11,12,37,23,28,18,15,19,62,38,47,29,

%T 24,16,32,103,63,78,48,39,27,20,53,172,104,129,79,64,44,33,21,88,287,

%U 173,214,132,107,73,54,34,25,147,478,288,357,219,178,122,89

%N Dispersion of A047202, (numbers >1 and congruent to 2 or 3 or 4 mod 5), by antidiagonals.

%C For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.

%C ...

%C Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}. Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5. There are 10 sequences in S, each matched by a (nearly) complementary sequence in T. Each of the 20 sequences generates a dispersion, as listed here:

%C ...

%C A191722=dispersion of A008851 (0, 1 mod 5 and >1)

%C A191723=dispersion of A047215 (0, 2 mod 5 and >1)

%C A191724=dispersion of A047218 (0, 3 mod 5 and >1)

%C A191725=dispersion of A047208 (0, 4 mod 5 and >1)

%C A191726=dispersion of A047216 (1, 2 mod 5 and >1)

%C A191727=dispersion of A047219 (1, 3 mod 5 and >1)

%C A191728=dispersion of A047209 (1, 4 mod 5 and >1)

%C A191729=dispersion of A047221 (2, 3 mod 5 and >1)

%C A191730=dispersion of A047211 (2, 4 mod 5 and >1)

%C A191731=dispersion of A047204 (3, 4 mod 5 and >1)

%C ...

%C A191732=dispersion of A047202 (2,3,4 mod 5 and >1)

%C A191733=dispersion of A047206 (1,3,4 mod 5 and >1)

%C A191734=dispersion of A032793 (1,2,4 mod 5 and >1)

%C A191735=dispersion of A047223 (1,2,3 mod 5 and >1)

%C A191736=dispersion of A047205 (0,3,4 mod 5 and >1)

%C A191737=dispersion of A047212 (0,2,4 mod 5 and >1)

%C A191738=dispersion of A047222 (0,2,3 mod 5 and >1)

%C A191739=dispersion of A008854 (0,1,4 mod 5 and >1)

%C A191740=dispersion of A047220 (0,1,3 mod 5 and >1)

%C A191741=dispersion of A047217 (0,1,2 mod 5 and >1)

%C ...

%C For further information about these 20 dispersions, see A191722.

%C ...

%C Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

%H Ivan Neretin, <a href="/A191732/b191732.txt">Table of n, a(n) for n = 1..5050</a>

%e Northwest corner:

%e 1....2....3....4....7

%e 5....8....13...22...37

%e 6....9....14...23...38

%e 10...17...28...47...78

%e 11...18...29...48...79

%e 15...24...39...64...107

%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=3; c2=4; m[n_]:=If[Mod[n,3]==0,1,0];

%t f[n_]:=a*m[n+2]+b*m[n+1]+c2*m[n]+5*Floor[(n-1)/3]

%t Table[f[n], {n, 1, 30}] (* A047202 *)

%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}]] (* A191732 *)

%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191732 *)

%Y Cf. A008851, A047202, A191722, A191426.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Jun 14 2011

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Last modified March 28 16:34 EDT 2024. Contains 371254 sequences. (Running on oeis4.)