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%I #4 Oct 05 2017 21:28:55
%S 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 40,28,13,32,47,57,61,59,51,35,15,37,56,69,77,80,73,60,41,18,43,66,84,
%U 94,101,97,88,71,49,19,50,76,99,113,123,124,115,103,82
%N 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 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 Clark Kimberling, <a href="/A292958/b292958.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=sqrt(5) and [ ]=floor.
%e Northwest corner:
%e 1 2 3 5 6 9 10 13 15
%e 4 7 11 16 20 26 32 37 43
%e 8 14 21 29 38 47 56 66 76
%e 12 22 33 44 57 69 84 99 112
%e 17 30 45 61 77 94 113 132 152
%e 24 40 59 80 101 123 146 169 194
%e 28 51 73 97 124 150 178 206 236
%e 35 60 88 115 147 180 212 247 282
%e The numbers k*(r+h), approximately:
%e (for k=1): 2.236 3.236 4.236 ...
%e (for k=2): 4.472 6.472 6.472 ...
%e (for k=3): 6.708 9.708 12.708 ...
%e Replacing each by its rank gives
%e 1 2 3
%e 4 7 11
%e 8 14 21
%t r = Sqrt[5]; 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] (* A292958 array *)
%t Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten (* A292958 sequence *)
%Y Cf. A182801.
%K nonn,easy,tabl
%O 1,2
%A _Clark Kimberling_, Oct 05 2017
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