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A292956 Rectangular array by antidiagonals: T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = sqrt(2), k>=1, h>=0, are jointly ranked.

%I A292956 #8 Aug 27 2022 21:30:51
%S A292956 1,2,3,4,7,5,6,11,13,9,8,17,21,19,12,10,23,30,32,26,16,14,29,39,46,44,
%T A292956 35,20,15,36,50,59,61,55,42,24,18,41,62,75,81,77,67,51,28,22,49,72,90,
%U A292956 100,102,95,82,60,33,25,56,84,106,120,128,125,113,93,69
%N A292956 Rectangular array by antidiagonals: T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = sqrt(2), k>=1, h>=0, are jointly ranked.
%C A292956 This is the transpose of the array at A182846.  Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers. [Sequence reference corrected by _Peter Munn_, Aug 27 2022]
%H A292956 Clark Kimberling, <a href="/A292956/b292956.txt">Antidiagonals n=1..60, flattened </a>
%F A292956 T(n,m) = Sum_{k=1...[n + m*n/r]} [1 - r + n*(r + m)/k], where r=sqrt(2) and [ ]=floor.
%e A292956 Northwest corner:
%e A292956 1   2    4    6    8    10   14   15   18
%e A292956 3   7    11   17   23   29   36   41   49
%e A292956 5   13   21   30   39   50   62   72   84
%e A292956 9   19   32   46   59   75   90   106  124
%e A292956 12  26   44   61   81   100  120  142  165
%e A292956 The numbers k*(r+h), approximately:
%e A292956 (for k=1):   1.414   2.414   3.414 ...
%e A292956 (for k=2):   2.828   4.828   6.828 ...
%e A292956 (for k=3):   4.242   7.242   10.242 ...
%e A292956 Replacing each by its rank gives
%e A292956 1   2    4
%e A292956 3   7    11
%e A292956 5   13   21
%t A292956 r = Sqrt[2]; z = 12;
%t A292956 t[n_, m_] := Sum[Floor[1 - r + n*(r + m)/k], {k, 1, Floor[n + m*n/r]}];
%t A292956 u = Table[t[n, m], {n, 1, z}, {m, 0, z}]; TableForm[u]  (* A292956 array *)
%t A292956 Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292956 sequence *)
%Y A292956 Cf. A182846.
%K A292956 nonn,easy,tabl
%O A292956 1,2
%A A292956 _Clark Kimberling_, Oct 04 2017