cp's OEIS Frontend

A191437 Dispersion of ([n*x+n+x-2]), where x=(golden ratio) and [ ]=floor, by antidiagonals.

%I A191437 #13 Oct 21 2024 00:47:05
%S A191437 1,3,2,8,5,4,21,13,11,6,55,34,29,16,7,144,89,76,42,18,9,377,233,199,
%T A191437 110,47,24,10,987,610,521,288,123,63,26,12,2584,1597,1364,754,322,165,
%U A191437 68,32,14,6765,4181,3571,1974,843,432,178,84,37,15,17711,10946
%N A191437 Dispersion of ([n*x+n+x-2]), where x=(golden ratio) and [ ]=floor, by antidiagonals.
%C A191437 Rows 1 and 2:  Fibonacci numbers.  Rows 3 and 5: Lucas numbers.  Row n satisfies the recurrence x(n)=3*x(n-1)-x(n-2).
%C A191437 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 A191437 (1) s=A000040 (the primes), D=A114537, u=A114538.
%C A191437 (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
%C A191437 (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
%C A191437 More recent examples of dispersions: A191426-A191455.
%e A191437 Northwest corner:
%e A191437   1....3....8....21...55
%e A191437   2....5....13...34...89
%e A191437   4....11...29...76...199
%e A191437   6....16...42...110..288
%e A191437   7....18...47...123..322
%t A191437 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191437 r = 40; r1 = 12;  c = 40; c1 = 12;  x = GoldenRatio;
%t A191437 f[n_] := Floor[n*x+n+x-2] (* complement of column 1 *)
%t A191437 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191437 rows = {NestList[f, 1, c]};
%t A191437 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191437 t[i_, j_] := rows[[i, j]];
%t A191437 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191437 (* A191437 array *)
%t A191437 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191437 sequence *)
%t A191437 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%Y A191437 Cf. A114537, A035513, A035506, A000045, A000032, A191426.
%K A191437 nonn,tabl
%O A191437 1,2
%A A191437 _Clark Kimberling_, Jun 04 2011