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A292959 Rectangular array by antidiagonals: T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = (1+sqrt(5))/2 (the golden ratio), k>=1, h>=0, are jointly ranked.

%I A292959 #6 Dec 11 2023 10:47:37
%S A292959 1,2,3,4,7,6,5,11,13,9,8,16,21,19,14,10,22,30,31,27,18,12,28,39,45,43,
%T A292959 36,23,15,34,50,57,61,56,44,26,17,40,60,73,79,78,68,52,32,20,47,70,87,
%U A292959 98,101,94,83,63,37,24,54,82,104,118,126,124,113,96,72
%N A292959 Rectangular array by antidiagonals:  T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = (1+sqrt(5))/2 (the golden ratio), k>=1, h>=0, are jointly ranked.
%C A292959 This is the transpose of the array at A182849.  Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.
%H A292959 Clark Kimberling, <a href="/A292959/b292959.txt">Antidiagonals n=1..60, flattened</a>
%F A292959 T(n,m) = Sum_{k=1...[n + m*n/r]} [1 - r + n*(r + m)/k], where r=GoldenRatio and [ ]=floor.
%e A292959 Northwest corner:
%e A292959 1    2    4    5     8     10    12    15
%e A292959 3    7    11   16    22    28    34    40
%e A292959 6    13   21   30    39    50    60    70
%e A292959 9    19   31   45    57    73    87    104
%e A292959 14   27   43   61    79    98    118   138
%e A292959 18   36   56   78    101   126   150   176
%e A292959 23   44   68   94    124   152   184   215
%e A292959 26   52   83   113   146   181   217   255
%e A292959 The numbers k*(r+h), approximately:
%e A292959 (for k=1):   1.618   2.618   3.618 ...
%e A292959 (for k=2):   3.236   5.236   7.236 ...
%e A292959 (for k=3):   4.854   7.854   10.854 ...
%e A292959 Replacing each by its rank gives
%e A292959 1     2      4
%e A292959 3     7      11
%e A292959 6     13     21
%t A292959 r = GoldenRatio; z = 12;
%t A292959 t[n_, m_] := Sum[Floor[1 - r + n*(r + m)/k], {k, 1, Floor[n + m*n/r]}];
%t A292959 u = Table[t[n, m], {n, 1, z}, {m, 0, z}]; TableForm[u]  (* A292959 array *)
%t A292959 Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292959 sequence *)
%Y A292959 Cf. A182801, A292960, A292961.
%K A292959 nonn,easy,tabl
%O A292959 1,2
%A A292959 _Clark Kimberling_, Oct 05 2017