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A292965 Rectangular array by antidiagonals: T(n,m) = rank of n*(Pi + m) when all the numbers k*(Pi+h), for k >= 1, h >= 0, are jointly ranked.

%I A292965 #6 Oct 06 2017 21:34:21
%S A292965 1,2,5,3,8,10,4,12,16,17,6,15,22,26,23,7,20,30,35,36,31,9,25,38,46,50,
%T A292965 47,39,11,29,45,58,64,65,59,48,13,34,54,70,78,84,79,71,56,14,41,63,83,
%U A292965 95,103,104,97,86,67,18,44,73,94,113,123,127,124,115,99
%N A292965 Rectangular array by antidiagonals: T(n,m) = rank of n*(Pi + m) when all the numbers k*(Pi+h), for k >= 1, h >= 0, are jointly ranked.
%C A292965 Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.
%H A292965 Clark Kimberling, <a href="/A292965/b292965.txt">Antidiagonals n=1..60, flattened</a>
%F A292965 T(n,m) = Sum_{k=1...[n + m*n/Pi]} [1 - Pi + n*(Pi + m)/k], where [ ]=floor.
%F A292965 Northwest corner:
%F A292965 1      2     3     4     6     7
%F A292965 5      8     12    15    20    25
%F A292965 10     16    22    30    38    45
%F A292965 17     26    35    46    58    70
%F A292965 23     36    50    64    78    95
%F A292965 31     47    65    84    103   123
%F A292965 39     59    79    104   127   153
%F A292965 The numbers k*(Pi+h), approximately:
%F A292965 (for k=1):   3.141   4.141   5.141 ...
%F A292965 (for k=2):   6.283   8.283   10.283 ...
%F A292965 (for k=3):   9.424   12.424  15.424 ...
%F A292965 Replacing each by its rank gives
%F A292965 1    2    3
%F A292965 5    8    12
%F A292965 10   16   22
%t A292965 r = Pi; z = 12;
%t A292965 t[n_, m_] := Sum[Floor[1 - r + n*(r + m)/k], {k, 1, Floor[n + m*n/r]}];
%t A292965 u = Table[t[n, m], {n, 1, z}, {m, 0, z}]; TableForm[u]  (* A292965 array *)
%t A292965 Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292965 sequence *)
%Y A292965 Cf. A182801.
%K A292965 nonn,easy,tabl
%O A292965 1,2
%A A292965 _Clark Kimberling_, Oct 06 2017