A292959 -id:A292959 - cp's OEIS frontend

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.

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

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   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

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

Cf. A182801, A292959, A292961.

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 *)

T(n,m) = Sum_{k=1...[n + m*n/r]} [1 - r + n*(r + m)/k], where r=(GoldenRatio)^2 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

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.

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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

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.

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Mathematica

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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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Mathematica

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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.

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.