A292960 -id:A292960 - cp's OEIS frontend

A292959 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 (the golden ratio), k>=1, h>=0, are jointly ranked.

1, 2, 3, 4, 7, 6, 5, 11, 13, 9, 8, 16, 21, 19, 14, 10, 22, 30, 31, 27, 18, 12, 28, 39, 45, 43, 36, 23, 15, 34, 50, 57, 61, 56, 44, 26, 17, 40, 60, 73, 79, 78, 68, 52, 32, 20, 47, 70, 87, 98, 101, 94, 83, 63, 37, 24, 54, 82, 104, 118, 126, 124, 113, 96, 72

Offset: 1

Clark Kimberling, Oct 05 2017

This is the transpose of the array at A182849. Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.

			Northwest corner:
1    2    4    5     8     10    12    15
3    7    11   16    22    28    34    40
6    13   21   30    39    50    60    70
9    19   31   45    57    73    87    104
14   27   43   61    79    98    118   138
18   36   56   78    101   126   150   176
23   44   68   94    124   152   184   215
26   52   83   113   146   181   217   255
The numbers k*(r+h), approximately:
(for k=1):   1.618   2.618   3.618 ...
(for k=2):   3.236   5.236   7.236 ...
(for k=3):   4.854   7.854   10.854 ...
Replacing each by its rank gives
1     2      4
3     7      11
6     13     21

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

Cf. A182801, A292960, A292961.

r = GoldenRatio; 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]  (* A292959 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292959 sequence *)

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

A292961 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.

Original entry on oeis.org

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, 33, 11, 24, 49, 66, 74, 72, 61, 41, 14, 28, 58, 82, 93, 98, 91, 73, 46, 16, 32, 67, 96, 115, 124, 122, 108, 85, 55, 18, 37, 78, 111, 136, 151, 155, 146, 129, 101

Offset: 1

Clark Kimberling, Oct 05 2017

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

			Northwest corner:
1    3    6    9    12   17   20
2    8    15   22   30   40   49
4    13   25   38   52   66   82
5    19   35   54   74   93   115
7    27   48   72   98   124  151
10   33   61   91   122  155  190
11   41   73   108  146  187  226
14   46   85   129  172  218  266
The numbers k*(r+h), approximately:
(for k=1):   0.618   1.618   2.618 ...
(for k=2):   1.236   3.236   5.236 ...
(for k=3):   1.854   4.854   7.854 ...
Replacing each k*(r+h) by its rank gives
1    3    6
2    8    15
4    13   25

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

Cf. A182801, A292959, A292960.

r = -1+GoldenRatio; 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]  (* A292961 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292961 sequence *)

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

cp's OEIS Frontend

A292959 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 (the golden ratio), k>=1, h>=0, are jointly ranked.

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A292961 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.

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cp's OEIS Frontend

A292959 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 (the golden ratio), k>=1, h>=0, are jointly ranked.

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Programs

Mathematica

Formula

A292961 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.

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Programs

Mathematica

Formula

A292959 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 (the golden ratio), k>=1, h>=0, are jointly ranked.

A292961 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.