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A191451 Dispersion of (3*n-2), for n>=2, by antidiagonals.

%I A191451 #19 Oct 21 2024 00:51:31
%S A191451 1,4,2,13,7,3,40,22,10,5,121,67,31,16,6,364,202,94,49,19,8,1093,607,
%T A191451 283,148,58,25,9,3280,1822,850,445,175,76,28,11,9841,5467,2551,1336,
%U A191451 526,229,85,34,12,29524,16402,7654,4009,1579,688,256,103,37,14,88573
%N A191451 Dispersion of (3*n-2), for n>=2, by antidiagonals.
%C A191451 Row 1:  A003462
%C A191451 Row 2:  A060816
%C A191451 Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
%C A191451 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191451 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191451 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191451 More recent examples of dispersions: A191426-A191455.
%e A191451 Northwest corner:
%e A191451   1...4....13...40...121
%e A191451   2...7....22...67...202
%e A191451   3...10...31...94...283
%e A191451   5...16...49...148..445
%e A191451   6...19...58...175..526
%t A191451 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191451 r=40; r1=12; c=40; c1=12;
%t A191451 f[n_] :=3n+1 (* complement of column 1 *)
%t A191451 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191451 rows = {NestList[f, 1, c]};
%t A191451 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191451 t[i_, j_] := rows[[i, j]];
%t A191451 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191451 (* A191451 array *)
%t A191451 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191451 sequence *)
%t A191451 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191451 Cf. A114537, A035513, A035506, A191449.
%K A191451 nonn,tabl
%O A191451 1,2
%A A191451 _Clark Kimberling_, Jun 05 2011