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A292963 Rectangular array by antidiagonals: T(n,m) = rank of n*(e + m) when all the numbers k*(e+h), for k>=1, h>=0, are jointly ranked.

%I A292963 #4 Oct 05 2017 21:29:32
%S A292963 1,2,4,3,7,9,5,11,15,14,6,16,22,24,20,8,19,29,34,32,27,10,25,38,45,48,
%T A292963 43,35,12,30,46,57,62,61,54,42,13,36,55,70,79,81,76,67,50,17,40,64,83,
%U A292963 95,101,100,92,78,58,18,47,73,97,113,122,125,120,108,89
%N A292963 Rectangular array by antidiagonals:  T(n,m) = rank of n*(e + m) when all the numbers k*(e+h), for k>=1, h>=0, are jointly ranked.
%C A292963 Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.
%H A292963 Clark Kimberling, <a href="/A292963/b292963.txt">Antidiagonals n=1..60, flattened</a>
%F A292963 T(n,m) = Sum_{k=1...[n + m*n/e]} [1 -  e + n*(e + m)/k], where [ ]=floor.
%e A292963 Northwest corner:
%e A292963 1    2    3     5   6    8
%e A292963 4    7    11   16   19   25
%e A292963 9    15   22   29   38   46
%e A292963 14   24   34   45   57   70
%e A292963 20   32   48   62   79   95
%e A292963 27   43   61   81   101  122
%e A292963 35   54   76   100  125  152
%e A292963 42   67   92   120  151  181
%e A292963 The numbers k*(r+h), approximately:
%e A292963 (for k=1):   2.718  3.718  4.718 ...
%e A292963 (for k=2):   5.436   7.436   9.436 ...
%e A292963 (for k=3):   8.154   11.854   14.154 ...
%e A292963 Replacing each by its rank gives
%e A292963 1    2    3
%e A292963 4    7    14
%e A292963 9    15   22
%t A292963 r = E; z = 12;
%t A292963 t[n_, m_] := Sum[Floor[1 - r + n*(r + m)/k], {k, 1, Floor[n + m*n/r]}];
%t A292963 u = Table[t[n, m], {n, 1, z}, {m, 0, z}]; TableForm[u]  (* A292963 array *)
%t A292963 Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292963 sequence *)
%Y A292963 Cf. A182801, A292964.
%K A292963 nonn,easy,tabl
%O A292963 1,2
%A A292963 _Clark Kimberling_, Oct 05 2017