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