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%I #6 Dec 11 2023 10:47:37
%S 1,2,3,4,7,6,5,11,13,9,8,16,21,19,14,10,22,30,31,27,18,12,28,39,45,43,
%T 36,23,15,34,50,57,61,56,44,26,17,40,60,73,79,78,68,52,32,20,47,70,87,
%U 98,101,94,83,63,37,24,54,82,104,118,126,124,113,96,72
%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 (the golden ratio), k>=1, h>=0, are jointly ranked.
%C This is the transpose of the array at A182849. Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.
%H Clark Kimberling, <a href="/A292959/b292959.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 and [ ]=floor.
%e Northwest corner:
%e 1 2 4 5 8 10 12 15
%e 3 7 11 16 22 28 34 40
%e 6 13 21 30 39 50 60 70
%e 9 19 31 45 57 73 87 104
%e 14 27 43 61 79 98 118 138
%e 18 36 56 78 101 126 150 176
%e 23 44 68 94 124 152 184 215
%e 26 52 83 113 146 181 217 255
%e The numbers k*(r+h), approximately:
%e (for k=1): 1.618 2.618 3.618 ...
%e (for k=2): 3.236 5.236 7.236 ...
%e (for k=3): 4.854 7.854 10.854 ...
%e Replacing each by its rank gives
%e 1 2 4
%e 3 7 11
%e 6 13 21
%t r = GoldenRatio; 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] (* A292959 array *)
%t Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten (* A292959 sequence *)
%Y Cf. A182801, A292960, A292961.
%K nonn,easy,tabl
%O 1,2
%A _Clark Kimberling_, Oct 05 2017
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