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