A292965 Rectangular array by antidiagonals: T(n,m) = rank of n*(Pi + m) when all the numbers k*(Pi+h), for k >= 1, h >= 0, are jointly ranked.

1, 2, 5, 3, 8, 10, 4, 12, 16, 17, 6, 15, 22, 26, 23, 7, 20, 30, 35, 36, 31, 9, 25, 38, 46, 50, 47, 39, 11, 29, 45, 58, 64, 65, 59, 48, 13, 34, 54, 70, 78, 84, 79, 71, 56, 14, 41, 63, 83, 95, 103, 104, 97, 86, 67, 18, 44, 73, 94, 113, 123, 127, 124, 115, 99

Offset: 1

Author

Clark Kimberling, Oct 06 2017

Keywords

• nonn
• easy
• tabl

Comments

Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.

Links

• Clark Kimberling, Antidiagonals n=1..60, flattened

Crossrefs

Cf. A182801.

Programs

Mathematica

r = Pi; 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]  (* A292965 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292965 sequence *)

Formula

T(n,m) = Sum_{k=1...[n + m*n/Pi]} [1 - Pi + n*(Pi + m)/k], where [ ]=floor.

Northwest corner:

1 2 3 4 6 7

5 8 12 15 20 25

10 16 22 30 38 45

17 26 35 46 58 70

23 36 50 64 78 95

31 47 65 84 103 123

39 59 79 104 127 153

The numbers k*(Pi+h), approximately:

(for k=1): 3.141 4.141 5.141 ...

(for k=2): 6.283 8.283 10.283 ...

(for k=3): 9.424 12.424 15.424 ...

Replacing each by its rank gives

1 2 3

5 8 12

10 16 22