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A292958 Rectangular array by antidiagonals: T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = sqrt(5), k>=1, h>=0, are jointly ranked.

%I A292958 #4 Oct 05 2017 21:28:55
%S A292958 1,2,4,3,7,8,5,11,14,12,6,16,21,22,17,9,20,29,33,30,24,10,26,38,44,45,
%T A292958 40,28,13,32,47,57,61,59,51,35,15,37,56,69,77,80,73,60,41,18,43,66,84,
%U A292958 94,101,97,88,71,49,19,50,76,99,113,123,124,115,103,82
%N A292958 Rectangular array by antidiagonals:  T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = sqrt(5), k>=1, h>=0, are jointly ranked.
%C A292958 This is the transpose of the array at A182848.  Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.
%H A292958 Clark Kimberling, <a href="/A292958/b292958.txt">Antidiagonals n=1..60, flattened</a>
%F A292958 T(n,m) = Sum_{k=1...[n + m*n/r]} [1 - r + n*(r + m)/k], where r=sqrt(5) and [ ]=floor.
%e A292958 Northwest corner:
%e A292958 1     2      3      5      6      9     10     13     15
%e A292958 4     7      11     16     20     26    32     37     43
%e A292958 8     14     21     29     38     47    56     66     76
%e A292958 12    22     33     44     57     69    84     99     112
%e A292958 17    30     45     61     77     94    113    132    152
%e A292958 24    40     59     80     101    123   146    169    194
%e A292958 28    51     73     97     124    150   178    206    236
%e A292958 35    60     88     115    147    180   212    247    282
%e A292958 The numbers k*(r+h), approximately:
%e A292958 (for k=1):   2.236   3.236   4.236 ...
%e A292958 (for k=2):   4.472   6.472   6.472 ...
%e A292958 (for k=3):   6.708   9.708   12.708 ...
%e A292958 Replacing each by its rank gives
%e A292958 1      2      3
%e A292958 4      7      11
%e A292958 8      14     21
%t A292958 r = Sqrt[5]; z = 12;
%t A292958 t[n_, m_] := Sum[Floor[1 - r + n*(r + m)/k], {k, 1, Floor[n + m*n/r]}];
%t A292958 u = Table[t[n, m], {n, 1, z}, {m, 0, z}]; TableForm[u]  (* A292958 array *)
%t A292958 Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292958 sequence *)
%Y A292958 Cf. A182801.
%K A292958 nonn,easy,tabl
%O A292958 1,2
%A A292958 _Clark Kimberling_, Oct 05 2017