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

1,2

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

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

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 *)

Crossrefs

Cf. A182801.

Sequence in context: A258406 A136189 A309557 * A302054 A081146 A261927

Adjacent sequences: A292962 A292963 A292964 * A292966 A292967 A292968

Keyword

nonn,easy,tabl

Author

Clark Kimberling, Oct 06 2017

Status

approved