A182847 -id:A182847 - cp's OEIS frontend

A182801 Joint-rank array of the numbers j*r^(i-1), where r = golden ratio = (1+sqrt(5))/2, i>=1, j>=1, read by antidiagonals.

1, 3, 2, 5, 6, 4, 7, 9, 11, 8, 10, 13, 16, 19, 14, 12, 18, 23, 28, 32, 25, 15, 21, 31, 39, 48, 54, 42, 17, 26, 36, 52, 66, 81, 89, 71, 20, 29, 44, 61, 86, 110, 134, 147, 117, 22, 34, 49, 73, 102, 141, 181, 221, 240, 193, 24, 38, 57, 82

Offset: 1

Clark Kimberling, Dec 04 2010

Joint-rank arrays are introduced here as follows.
Suppose that R={f(i,j)} is set of positive numbers, where i and j range through countable sets I and J, respectively, such that for every n, then number f(i,j) < n is finite.  Let T(i,j) be the position of f(i,j) in the joint ranking of all the numbers in R.  The joint-rank array of R is the array T whose i-th row is T(i,j).
For A182801, f(i,j)=j*r^(i-1), where r=(1+sqrt(5))/2 and I=J={1,2,3,...}.
(row 1)=A020959; (row 2)=A020960; (row 3)=A020961.
(col 1)=A020956; (col 2)=A020957; (col 3)=A020958.
Every positive integer occurs exactly once in A182801, so that as a sequence it is a permutation of the positive integers.

			Northwest corner:
1....3....5....7...10...12...
2....6....9...13...18...21...
4...11...16...23...31...36...
8...19...28...39...52...61...

Cf. A001622, A182802, A182846, A182847, A182848, A182849, A252229.

r=GoldenRatio;
f[i_,j_]:=Sum[Floor[j*r^(i-k)],{k,1,i+Log[r,j]}];
TableForm[Table[f[i,j],{i,1,16},{j,1,16}]] (* A182801 *)

T(i,j)=Sum{floor(j*r^(i-k)): k>=1}.

A182846 Joint-rank array of the numbers j*(i-1+r), where r=sqrt(2), i>=1, j>=1, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 5, 7, 4, 9, 13, 11, 6, 12, 19, 21, 17, 8, 16, 26, 32, 30, 23, 10, 20, 35, 44, 46, 39, 29, 14, 24, 42, 55, 61, 59, 50, 36, 15, 28, 51, 67, 77, 81, 75, 62, 41, 18, 33, 60, 82, 95, 102, 100, 90, 72, 49, 22, 38, 69, 93, 113, 125, 128, 120, 106, 84, 56, 25, 43

Offset: 1

Clark Kimberling, Dec 08 2010

Joint-rank arrays are defined in the first comment at A182801.

			Northwest corner:
1....3....5....9...12...
2....7...13...19...26...
4...11...21...32...44...
6...17...30...46...61...
The numbers j*(i-1+sqrt(2)), approximately:
(for i=1)  1.41, 2.83, 4.24,...
(for i=2)  2.41, 4.83, 7.24,...
(for i=3)  3.41, 6.83, 10.24,...
Replacing each by its rank gives
1....3....5
2....7...13
4...ll...21

Cf. A182801, A182847-A182849.

r=Sqrt[2];
f[i_,j_]:=Sum[Floor[j*(i-1+r)/(k-1+r)],{k,1,1+r+j(i-1+r)}];
TableForm[Table[f[i,j],{i,1,10},{j,1,10}]] (*A182846*)

T(i,j)=SUM{floor(j*(i-1+r)/(k-1+r)): r=sqrt(2), k>=1} for i>=1, j>=1.

A292957 Rectangular array by antidiagonals: T(n,m) = rank of n(r+m) when all the numbers k(r+h), where r = sqrt(3), k>=1, h>=0, are jointly ranked.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 5, 11, 13, 10, 8, 16, 21, 20, 14, 9, 22, 30, 32, 27, 18, 12, 26, 38, 44, 42, 36, 24, 15, 33, 49, 58, 61, 55, 46, 29, 17, 40, 59, 72, 78, 77, 69, 54, 34, 19, 47, 70, 87, 98, 100, 95, 84, 64, 39, 23, 52, 80, 103, 117, 124, 123, 113, 97, 73

Offset: 1

Clark Kimberling, Oct 05 2017

This is the transpose of the array at A182847. 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     9     12    15
3    7     11    16    22    26    33    40
6    13    21    30    38    49    59    70
10   20    32    44    58    72    87    103
14   27    42    61    78    98    117   137
18   36    55    77    100   124   147   175
24   46    69    95    123   152   183   212
The numbers k*(r+h), approximately:
(for k=1):   1.732   2.732    3.732 ...
(for k=2):   3.464   5.464    7.464 ...
(for k=3):   5.196   8.196    12.296 ...
Replacing each by its rank gives
1    2     4
3    7     11
6    13    21

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

Cf. A182801, A182847.

r = Sqrt[3]; 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]  (* A292957 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292957 sequence *)

T(n,m) = Sum_{k=1...[n + m*n/r]} [1 - r + n*(r + m)/k], where r=sqrt(3) and [ ]=floor.

cp's OEIS Frontend

A182801 Joint-rank array of the numbers j*r^(i-1), where r = golden ratio = (1+sqrt(5))/2, i>=1, j>=1, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A182846 Joint-rank array of the numbers j*(i-1+r), where r=sqrt(2), i>=1, j>=1, by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A292957 Rectangular array by antidiagonals: T(n,m) = rank of n(r+m) when all the numbers k(r+h), where r = sqrt(3), k>=1, h>=0, are jointly ranked.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A182801 Joint-rank array of the numbers j*r^(i-1), where r = golden ratio = (1+sqrt(5))/2, i>=1, j>=1, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A182846 Joint-rank array of the numbers j*(i-1+r), where r=sqrt(2), i>=1, j>=1, by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A292957 Rectangular array by antidiagonals: T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = sqrt(3), k>=1, h>=0, are jointly ranked.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A292957 Rectangular array by antidiagonals: T(n,m) = rank of n(r+m) when all the numbers k(r+h), where r = sqrt(3), k>=1, h>=0, are jointly ranked.