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A292960 Rectangular array by antidiagonals: T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = ((1+sqrt(5))/2)^2, k>=1, h>=0, are jointly ranked.

%I A292960 #6 Dec 11 2023 10:47:25
%S A292960 1,2,4,3,7,9,5,11,15,13,6,16,22,23,19,8,20,29,34,32,27,10,25,38,44,47,
%T A292960 43,33,12,30,46,57,62,61,53,40,14,36,55,69,78,81,75,66,49,17,41,65,83,
%U A292960 95,102,100,91,76,56,18,48,74,96,112,122,124,119,107,88
%N A292960 Rectangular array by antidiagonals:  T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = ((1+sqrt(5))/2)^2, k>=1, h>=0, are jointly ranked.
%C A292960 Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.
%H A292960 Clark Kimberling, <a href="/A292960/b292960.txt">Antidiagonals n=1..60, flattened</a>
%F A292960 T(n,m) = Sum_{k=1...[n + m*n/r]} [1 - r + n*(r + m)/k], where r=(GoldenRatio)^2 and [ ]=floor.
%e A292960 Northwest corner:
%e A292960 1   2   3    5    6    8   10
%e A292960 4   7   11   16   20   25  30
%e A292960 9   15  22   29   38   46  55
%e A292960 13  23  34   44   57   69  83
%e A292960 19  32  47   62   78   95  112
%e A292960 27  43  61   81   102  122 145
%e A292960 The numbers k*(r+h), approximately:
%e A292960 (for k=1):   2.618   3.618   4.618 ...
%e A292960 (for k=2):   5.236   7.236   9.236 ...
%e A292960 (for k=3):   7.854   10.854   13.854 ...
%e A292960 Replacing each k*(r+h) by its rank gives
%e A292960 1    2     3
%e A292960 4    7     11
%e A292960 9    15    22
%t A292960 r = GoldenRatio^2; z = 12;
%t A292960 t[n_, m_] := Sum[Floor[1 - r + n*(r + m)/k], {k, 1, Floor[n + m*n/r]}];
%t A292960 u = Table[t[n, m], {n, 1, z}, {m, 0, z}]; TableForm[u]  (* A292960 array *)
%t A292960 Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292960 sequence *)
%Y A292960 Cf. A182801, A292959, A292961.
%K A292960 nonn,easy,tabl
%O A292960 1,2
%A A292960 _Clark Kimberling_, Oct 05 2017