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A191455 Dispersion of (floor(n*e)), by antidiagonals.

%I A191455 #14 Oct 21 2024 00:55:44
%S A191455 1,2,3,5,8,4,13,21,10,6,35,57,27,16,7,95,154,73,43,19,9,258,418,198,
%T A191455 116,51,24,11,701,1136,538,315,138,65,29,12,1905,3087,1462,856,375,
%U A191455 176,78,32,14,5178,8391,3974,2326,1019,478,212,86,38,15,14075,22809
%N A191455 Dispersion of (floor(n*e)), by antidiagonals.
%C A191455 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 A191455 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191455 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191455 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191455 More recent examples of dispersions: A191426-A191455.
%e A191455 Northwest corner:
%e A191455   1...2....5....13...35
%e A191455   3...8....21...57...154
%e A191455   4...10...27...73...198
%e A191455   6...16...43...116..315
%e A191455   7...19...51...138..375
%p A191455 A191455 := proc(r, c)
%p A191455     option remember;
%p A191455     if c = 1 then
%p A191455         A054385(r) ;
%p A191455     else
%p A191455         A022843(procname(r, c-1)) ;
%p A191455     end if;
%p A191455 end proc: # _R. J. Mathar_, Jan 25 2015
%t A191455 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191455 r=40; r1=12; c=40; c1=12;
%t A191455 f[n_] :=Floor[n*E]   (* complement of column 1 *)
%t A191455 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191455 rows = {NestList[f, 1, c]};
%t A191455 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191455 t[i_, j_] := rows[[i, j]];
%t A191455 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191455 (* A191455 array *)
%t A191455 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191455 sequence *)
%t A191455 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191455 Cf. A114537, A035513, A035506.
%K A191455 nonn,tabl
%O A191455 1,2
%A A191455 _Clark Kimberling_, Jun 05 2011