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Rectangular array by antidiagonals: T(n,m) = rank of n*sqrt(5)+m when all the numbers k*sqrt(5)+h, for k >= 1, h >= 0, are jointly ranked.
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%I #6 Oct 06 2017 21:34:38

%S 1,2,4,3,6,9,5,8,12,16,7,11,15,20,25,10,14,19,24,30,37,13,18,23,29,35,

%T 43,51,17,22,28,34,41,49,58,67,21,27,33,40,47,56,65,75,85,26,32,39,46,

%U 54,63,73,83,94,106,31,38,45,53,61,71,81,92,103,116,129

%N Rectangular array by antidiagonals: T(n,m) = rank of n*sqrt(5)+m when all the numbers k*sqrt(5)+h, for 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. As an array, this is the interspersion of sqrt(1/5); see A283962.

%H Clark Kimberling, <a href="/A293054/b293054.txt">Antidiagonals n=1..60, flattened</a>

%F T(n,m) = Sum_{k=1...n + [m/r]} m+1+[(n-k)r], where r = sqrt(5), [ ]=floor.

%e Northwest corner:

%e 1 2 3 5 7 10 13

%e 4 6 8 11 14 18 22

%e 9 12 15 19 23 28 33

%e 16 20 24 29 34 40 46

%e 25 30 35 41 47 54 61

%e 37 43 49 56 63 71 79

%e 51 58 65 73 81 90 99

%e 67 75 83 92 101 111 121

%e 85 94 103 113 123 134 145

%e The numbers k*r+h, approximately:

%e (for k=1): 2.236 3.236 3.236 ...

%e (for k=2): 4.472 5.472 6.472 ...

%e (for k=3): 6.708 7.708 8.708 ...

%e Replacing each k*r+h by its rank gives

%e 1 2 3

%e 4 6 8

%e 9 12 15

%t r = Sqrt[5]; z = 12;

%t t[n_, m_] := Sum[Floor[1 + m + (n - k) r], {k, 1, n + Floor[m/r]}];

%t u = Table[t[n, m], {n, 1, z}, {m, 0, z}]

%t Grid[u] (* A293054 array *)

%t Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten (* A293054 sequence *)

%Y Cf. A283962.

%K nonn,easy,tabl

%O 1,2

%A _Clark Kimberling_, Oct 06 2017