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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.
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%I #6 Oct 06 2017 21:34:21

%S 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 47,39,11,29,45,58,64,65,59,48,13,34,54,70,78,84,79,71,56,14,41,63,83,

%U 95,103,104,97,86,67,18,44,73,94,113,123,127,124,115,99

%N 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 Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.

%H Clark Kimberling, <a href="/A292965/b292965.txt">Antidiagonals n=1..60, flattened</a>

%F T(n,m) = Sum_{k=1...[n + m*n/Pi]} [1 - Pi + n*(Pi + m)/k], where [ ]=floor.

%F Northwest corner:

%F 1 2 3 4 6 7

%F 5 8 12 15 20 25

%F 10 16 22 30 38 45

%F 17 26 35 46 58 70

%F 23 36 50 64 78 95

%F 31 47 65 84 103 123

%F 39 59 79 104 127 153

%F The numbers k*(Pi+h), approximately:

%F (for k=1): 3.141 4.141 5.141 ...

%F (for k=2): 6.283 8.283 10.283 ...

%F (for k=3): 9.424 12.424 15.424 ...

%F Replacing each by its rank gives

%F 1 2 3

%F 5 8 12

%F 10 16 22

%t r = Pi; 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] (* A292965 array *)

%t Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten (* A292965 sequence *)

%Y Cf. A182801.

%K nonn,easy,tabl

%O 1,2

%A _Clark Kimberling_, Oct 06 2017