cp's OEIS Frontend

A191435 Dispersion of ([n*x+n+x]), where x=(golden ratio) and [ ]=floor, by antidiagonals.

%I A191435 #12 Oct 20 2024 15:59:46
%S A191435 1,5,2,15,7,3,41,20,10,4,109,54,28,13,6,287,143,75,36,18,8,753,376,
%T A191435 198,96,49,23,9,1973,986,520,253,130,62,26,11,5167,2583,1363,664,342,
%U A191435 164,70,31,12,13529,6764,3570,1740,897,431,185,83,34,14,35421,17710
%N A191435 Dispersion of ([n*x+n+x]), where x=(golden ratio) and [ ]=floor, by antidiagonals.
%C A191435 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 A191435 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191435 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191435 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191435 More recent examples of dispersions: A191426-A191455.
%e A191435 Northwest corner:
%e A191435   1....5....15...41...109
%e A191435   2....7....20...54...143
%e A191435   3....10...28...75...198
%e A191435   4....13...36...96...253
%e A191435   6....18...49...130..342
%t A191435 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191435 r = 40; r1 = 12;  c = 40; c1 = 12;  x = 1 + GoldenRatio;
%t A191435 f[n_] := Floor[n*x + x] (* f(n),complement of column 1 *)
%t A191435 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191435 rows = {NestList[f, 1, c]};
%t A191435 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191435 t[i_, j_] := rows[[i, j]];
%t A191435 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191435 (* A191435 array *)
%t A191435 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191435 sequence *)
%t A191435 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191435 Cf. A114537, A035513, A035506.
%K A191435 nonn,tabl
%O A191435 1,2
%A A191435 _Clark Kimberling_, Jun 04 2011