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A292960
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.
3
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, 43, 33, 12, 30, 46, 57, 62, 61, 53, 40, 14, 36, 55, 69, 78, 81, 75, 66, 49, 17, 41, 65, 83, 95, 102, 100, 91, 76, 56, 18, 48, 74, 96, 112, 122, 124, 119, 107, 88
OFFSET
1,2
COMMENTS
Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.
LINKS
FORMULA
T(n,m) = Sum_{k=1...[n + m*n/r]} [1 - r + n*(r + m)/k], where r=(GoldenRatio)^2 and [ ]=floor.
EXAMPLE
Northwest corner:
1 2 3 5 6 8 10
4 7 11 16 20 25 30
9 15 22 29 38 46 55
13 23 34 44 57 69 83
19 32 47 62 78 95 112
27 43 61 81 102 122 145
The numbers k*(r+h), approximately:
(for k=1): 2.618 3.618 4.618 ...
(for k=2): 5.236 7.236 9.236 ...
(for k=3): 7.854 10.854 13.854 ...
Replacing each k*(r+h) by its rank gives
1 2 3
4 7 11
9 15 22
MATHEMATICA
r = GoldenRatio^2; z = 12;
t[n_, m_] := Sum[Floor[1 - r + n*(r + m)/k], {k, 1, Floor[n + m*n/r]}];
u = Table[t[n, m], {n, 1, z}, {m, 0, z}]; TableForm[u] (* A292960 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten (* A292960 sequence *)
CROSSREFS
KEYWORD
nonn,easy,tabl
AUTHOR
Clark Kimberling, Oct 05 2017
STATUS
approved