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%I #6 Dec 11 2023 10:46:51
%S 1,3,2,6,8,4,9,15,13,5,12,22,25,19,7,17,30,38,35,27,10,20,40,52,54,48,
%T 33,11,24,49,66,74,72,61,41,14,28,58,82,93,98,91,73,46,16,32,67,96,
%U 115,124,122,108,85,55,18,37,78,111,136,151,155,146,129,101
%N Rectangular array by antidiagonals: T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = -1+(1+sqrt(5))/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="/A292961/b292961.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=1/GoldenRatio and [ ]=floor.
%e Northwest corner:
%e 1 3 6 9 12 17 20
%e 2 8 15 22 30 40 49
%e 4 13 25 38 52 66 82
%e 5 19 35 54 74 93 115
%e 7 27 48 72 98 124 151
%e 10 33 61 91 122 155 190
%e 11 41 73 108 146 187 226
%e 14 46 85 129 172 218 266
%e The numbers k*(r+h), approximately:
%e (for k=1): 0.618 1.618 2.618 ...
%e (for k=2): 1.236 3.236 5.236 ...
%e (for k=3): 1.854 4.854 7.854 ...
%e Replacing each k*(r+h) by its rank gives
%e 1 3 6
%e 2 8 15
%e 4 13 25
%t r = -1+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] (* A292961 array *)
%t Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten (* A292961 sequence *)
%Y Cf. A182801, A292959, A292960.
%K nonn,easy,tabl
%O 1,2
%A _Clark Kimberling_, Oct 05 2017
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