cp's OEIS Frontend

A333029 Order array of array A333028, read by antidiagonals.

%I A333029 #10 Feb 05 2021 18:18:59
%S A333029 1,2,4,3,6,10,5,8,15,11,7,12,24,18,13,9,19,38,28,21,16,14,30,58,45,32,
%T A333029 25,17,22,47,77,65,50,40,27,20,35,67,95,84,70,60,44,31,23,55,86,112,
%U A333029 102,89,79,64,49,36,26,74,104,128,118,106,97,83,69,56,43
%N A333029 Order array of array A333028, read by antidiagonals.
%C A333029 Suppose that (r(n,k)), for n >=1 and k >= 1 is an array of distinct numbers.  Replace each r(n,k) by its position when all the numbers r(n,k) are ordered by <.  The resulting array is the order array of (r(n,k)).  The array A333029 is an interspersion and, as a sequence, a permutation of the natural numbers.
%D A333029 Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168.
%H A333029 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Kimberling/kimber12.html">Lucas Representations of Positive Integers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.
%e A333029 Northwest corner:
%e A333029    1   2   3   5   7    9   14   22
%e A333029    4   6   8  12  19   30   47   67
%e A333029   10  15  24  38  58   77   95  112
%e A333029   11  18  24  45  65   84  102  118
%e A333029   13  21  32  50  70   89  106  122
%e A333029   16  25  40  60  79   97  114  130
%e A333029   17  27  44  64  83  101  117  132
%t A333029 W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
%t A333029 t = Table[GCD[W[n, 1], W[n, 2]], {n, 1, 500}];
%t A333029 u = Flatten[Position[t, 1]] ; v[n_, k_] := W[u[[n]], k];
%t A333029 g = Sort[Table[v[n - k + 1, k], {n, 20}, {k, n, 1, -1}] // Flatten]
%t A333029 wo[n_, k_] := Length[Intersection[Range[v[n, k]], Complement[Range[1500], g]]]
%t A333029 WP[n_, k_] := v[n, k] - wo[n, k];
%t A333029 TableForm[Table[WP[n, k], {n, 1, 15}, {k, 1, 10}]]  (* A333029 array *)
%t A333029 Table[WP[n - k + 1, k], {n, 16}, {k, n, 1, -1}] // Flatten  (* A333029 sequence *)
%Y A333029 Cf. A333028.
%K A333029 nonn,tabl
%O A333029 1,2
%A A333029 _Clark Kimberling_, Mar 10 2020