A182801 -id:A182801 - 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.

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.

Original entry on oeis.org

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.

A178431 Joint-rank array of the sums of two primes.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 6, 5, 5, 6, 9, 7, 7, 7, 9, 11, 10, 8, 8, 10, 11, 14, 12, 12, 10, 12, 12, 14, 16, 15, 13, 13, 13, 13, 15, 16, 19, 17, 17, 15, 17, 15, 17, 17, 19, 23, 20, 18, 18, 18, 18, 18, 18, 20, 23, 25, 24, 21, 20, 21, 20, 21, 20, 21, 24, 25

Offset: 1

Clark Kimberling, Dec 21 2010

Joint-rank arrays are defined at A157927 for arrays in which duplicates occur (and otherwise, at A182801).

			A corner of the array A178400 of sums of two primes:
4...5...7...9..13...
5...6...8..10..14...
7...8..10..12..14...
9..10..12..14..18...
Each of these P(i,j) is replaced by its rank when all the
numbers P(h,k) are jointly ranked, leaving A178431:
1...2...4...6...9...
2...3...5...7..10...
4...5...7...8..12...
6...7...8..10..13...
The number of distinct sums p+q <=13 is 9.

Cf. A014092, A157927, A178400.

A182802 Joint-rank array of the numbers j(1+tau)^2(i-1), where tau = golden ratio = (1+sqrt(5))/2, i>=1, read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 5, 11, 20, 26, 6, 15, 31, 55, 72, 8, 19, 42, 84, 148, 194, 10, 23, 53, 113, 224, 393, 515, 12, 28, 64, 142, 300, 592, 1036, 1357, 13, 32, 75, 171, 376, 791, 1557, 2721, 3563, 14, 36, 86, 200, 452, 990, 2078, 4085, 7134, 9340

Offset: 1

Clark Kimberling, Dec 05 2010

Joint-rank arrays are defined in the first comment at A182801. There r=tau; here r=1+tau=(tau)^2.

(Column 1)=A054963. Every positive integer occurs exactly once, so that as a sequence, A182802 is a permutation of the positive integers.

			Northwest corner:
1....2....4....5....6...
3....7...11...15...19...
9...20...31...42...53...
26..55...84..113..142...

Cf. A182801.

T(i,j)=SUM{floor(j*(tau)^(2*(i-n))): n>=1}.

A182831 Joint-rank array of numbers j*r^(i-1), where r=1+sqrt(2), read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 11, 17, 22, 7, 14, 28, 45, 55, 9, 19, 37, 70, 112, 137, 10, 23, 48, 93, 171, 276, 334, 12, 26, 57, 118, 228, 417, 671, 812, 13, 31, 66, 141, 287, 556, 1010, 1627, 1965, 15, 34, 77, 164, 344, 697, 1347, 2444, 3934, 4751, 16, 39

Offset: 1

Clark Kimberling, Dec 07 2010

Joint-rank arrays are defined in the first comment at A182801. (row 1)=A087063. First 3 columns are A020062, A020063, A020064.

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 ...
   3  6 11 14 ...
   8 17 28 37 ...
  22 45 70 93 ...
  ...

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A182801.

T[n_, k_] := Sum[Floor[k*(1 + Sqrt[2])^(n - j)], {j, 1, 100}]; Table[T[k + 1, n - k], {n,1,10}, {k, 0, n-1}]//Flatten (* G. C. Greubel, Aug 18 2018 *)

T(i,j) = Sum_{n>=1} floor(j*(1+sqrt(2))^(i-n)).

A182832 Joint-rank array of numbers j*r^(i-1), where r=1+sqrt(3), read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 5, 12, 21, 30, 6, 15, 34, 61, 85, 8, 19, 44, 95, 172, 237, 9, 24, 56, 125, 262, 476, 652, 11, 28, 68, 157, 347, 718, 1307, 1788, 13, 32, 80, 190, 435, 955, 1965, 3579, 4891, 14, 37, 91, 222, 524, 1196, 2618, 5373, 9786, 13371, 16, 41, 104

Offset: 1

Clark Kimberling, Dec 07 2010

Joint-rank arrays are defined in the first comment at A182801. Every positive integer occurs exactly once, so that as a sequence A182832 is a permutation of the positive integers.

			Northwest corner:
1....2....4....5....6...
3....7...12...15...19...
10..21...34...44...56...
30..61...95..125..157...

Cf. A182801.

T(i,j)=SUM{floor(j*(1+sqrt(3))^(i-n)): n>=1}.

A182949 Joint-rank array of the numbers (3i+1)3^j, where i>=0, j>=0, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 7, 4, 14, 19, 11, 6, 41, 55, 32, 16, 8, 122, 163, 95, 46, 21, 9, 365, 487, 284, 136, 60, 25, 10, 1094, 1459, 851, 406, 177, 73, 29, 12, 3281, 4375, 2552, 1216, 528, 217, 86, 34, 13, 9842, 13123, 7655, 3646, 1581, 649, 257, 100, 38, 15

Offset: 1

Clark Kimberling, Dec 15 2010

Joint-rank arrays are defined in the first comment at A182801.  As for any joint-rank array, A182949 is a permutation of the positive integers, but, a fortiori, A182949 is an interspersion: after initial terms every row is interspersed with all other rows.  The numbers (3*i+1)*3^j as an array comprise A182828; and sorted, A026225.
(row 1)=A007051.
(row 2)=A052919.
(col 1)=A182829.

			Northwest corner:
1....2....5....14...
3....7...19....55...
4...11...32....95...
6...16...46...136...

Clark Kimberling, Interspersions and Dispersions.

Cf. A182801, A182828, A026225, A182950.

 M[i_,j_]:=j+Floor[Log[3*i+1]/Log[3]]; T[i_,j_]:=Sum[Floor[2/3+(3*i+1)*3^(j-k-1)],{k,0,M[i,j]}]; TableForm[Table[T[i,j],{i,0,9},{j,0,9}]]

A182950 Joint-rank array of the numbers (3i+2)3^j, where i>=0, j>=0, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 9, 7, 4, 27, 22, 12, 5, 81, 67, 36, 16, 6, 243, 202, 108, 49, 20, 8, 729, 607, 324, 148, 62, 25, 10, 2187, 1822, 972, 445, 188, 76, 30, 11, 6561, 5467, 2916, 1336, 566, 229, 90, 34, 13, 19683, 16402, 8748, 4009, 1700, 688, 270, 103, 39, 14

Offset: 1

Clark Kimberling, Dec 15 2010

Joint-rank arrays are defined in the first comment at A182801.  As for any joint-rank array, A182950 is a permutation of the positive integers, but, a fortiori, A182950 is an interspersion: after initial terms every row is interspersed with all other rows.  The numbers (3*i+2)*3^j as an array comprise A182830; and sorted, possibly A026179.
(row 1)=A000244.
(row 2)=A060816.
(row 3)=A003946.
(row 4)=A052909.
(row 5)=A027107?

			Northwest corner:
1....3....9....27...
2....7...22....67...
4...12...36...108...
5...16...49...148...

Clark Kimberling, Interspersions and Dispersions.

Cf. A182801, A182830, A026179, A182949.

 M[i_,j_]:=j+Floor[Log[3*i/2+1]/Log[3]];
 T[i_,j_]:=Sum[Floor[1/3+(3*i+2)*3^(j-k-1)],{k,0,M[i,j]}];
 TableForm[Table[T[i,j],{i,0,9},{j,0,9}]]

A252229 The number of numbers j*r^k in the interval [n,n+1), where r = (1 + sqrt(5))/2, the golden ratio, and j >=0, k >= 0.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3, 2, 2, 3, 2, 3, 2, 4, 2, 2, 2, 4, 3, 3, 2, 2, 3, 2, 2, 3, 3, 2, 3, 2, 4, 3, 2, 2, 3, 2, 2, 3, 3, 4, 2, 2, 3, 3, 2, 3, 2, 3, 2, 2, 3, 3, 3, 2, 2, 3, 3, 2, 2, 3, 4, 3, 2, 2, 3, 2, 3, 2, 3, 2

Offset: 0

Clark Kimberling, Dec 16 2014

The least n for which a(n) = 4 is 29; the least n for which a(n) = 5 is 199.

			in [0,1):  0
in [1,2):  1, 1 + r
in [2,3):  2, 2 + r
in [3,4):  3, 1+2*r
in [4,5):  4, 1+3*r, 2 + r

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A182801, A020959.

z = 100; r = (1 + Sqrt[5])/2;
s[n_, j_] := s[n, j] = Floor[Log[n/j]/Log[r]];
a[n_] := a[n] = Sum[s[n + 1, j] - s[n, j], {j, 1, Floor[(n + 1)/r]}];
t = Join[{1}, Table[1 + a[n], {n, 1, z}]] (* A252229 *)

a(n) = 1 + sum{s(n+1,j) - s(n,j), j=1..floor[(n+1)/r]}, where s(n,j) = floor[log(n/j)/log(r)], for n >= 1.

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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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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A178431 Joint-rank array of the sums of two primes.

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A182802 Joint-rank array of the numbers j*(1+tau)^2*(i-1), where tau = golden ratio = (1+sqrt(5))/2, i>=1, read by antidiagonals.

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A182831 Joint-rank array of numbers j*r^(i-1), where r=1+sqrt(2), read by antidiagonals.

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A182832 Joint-rank array of numbers j*r^(i-1), where r=1+sqrt(3), read by antidiagonals.

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A182949 Joint-rank array of the numbers (3*i+1)*3^j, where i>=0, j>=0, by antidiagonals.

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A182950 Joint-rank array of the numbers (3*i+2)*3^j, where i>=0, j>=0, by antidiagonals.

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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.

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.

A182802 Joint-rank array of the numbers j(1+tau)^2(i-1), where tau = golden ratio = (1+sqrt(5))/2, i>=1, read by antidiagonals.

A182949 Joint-rank array of the numbers (3i+1)3^j, where i>=0, j>=0, by antidiagonals.

A182950 Joint-rank array of the numbers (3i+2)3^j, where i>=0, j>=0, by antidiagonals.