A182848 -id:A182848 - 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}.

A182847 Joint-rank array of the numbers j*(i-1+r), where r=sqrt(3), i>=1, j>=1, by diagonals.

Original entry on oeis.org

1, 3, 2, 6, 7, 4, 10, 13, 11, 5, 14, 20, 21, 16, 8, 18, 27, 32, 30, 22, 9, 24, 36, 42, 44, 38, 26, 12, 29, 46, 55, 61, 58, 49, 33, 15, 34, 54, 69, 77, 78, 72, 59, 40, 17, 39, 64, 84, 95, 100, 98, 87, 70, 47, 19, 45, 73, 97, 113, 123, 124, 117, 103, 80

Offset: 1

Clark Kimberling, Dec 08 2010

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

			Northwest corner:
1....3....6...10...
2....7...13...20...
4...11...21...32...
5...16...30...44...

Cf. A182801, A182846, A182848, A182849.

r=Sqrt[3];
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}]]

T(i,j)=sum(k>=1, floor( j*(i-1+r)/(k-1+r) ) ) where r=sqrt(3), for i>=1, j>=1.

A292958 Rectangular array by antidiagonals: T(n,m) = rank of n(r+m) when all the numbers k(r+h), where r = sqrt(5), k>=1, h>=0, are jointly ranked.

Original entry on oeis.org

1, 2, 4, 3, 7, 8, 5, 11, 14, 12, 6, 16, 21, 22, 17, 9, 20, 29, 33, 30, 24, 10, 26, 38, 44, 45, 40, 28, 13, 32, 47, 57, 61, 59, 51, 35, 15, 37, 56, 69, 77, 80, 73, 60, 41, 18, 43, 66, 84, 94, 101, 97, 88, 71, 49, 19, 50, 76, 99, 113, 123, 124, 115, 103, 82

Offset: 1

Clark Kimberling, Oct 05 2017

This is the transpose of the array at A182848. 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      9     10     13     15
4     7      11     16     20     26    32     37     43
8     14     21     29     38     47    56     66     76
12    22     33     44     57     69    84     99     112
17    30     45     61     77     94    113    132    152
24    40     59     80     101    123   146    169    194
28    51     73     97     124    150   178    206    236
35    60     88     115    147    180   212    247    282
The numbers k*(r+h), approximately:
(for k=1):   2.236   3.236   4.236 ...
(for k=2):   4.472   6.472   6.472 ...
(for k=3):   6.708   9.708   12.708 ...
Replacing each by its rank gives
1      2      3
4      7      11
8      14     21

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

Cf. A182801.

r = Sqrt[5]; 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]  (* A292958 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292958 sequence *)

T(n,m) = Sum_{k=1...[n + m*n/r]} [1 - r + n*(r + m)/k], where r=sqrt(5) 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.

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A182847 Joint-rank array of the numbers j*(i-1+r), where r=sqrt(3), i>=1, j>=1, by diagonals.

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A292958 Rectangular array by antidiagonals: T(n,m) = rank of n(r+m) when all the numbers k(r+h), where r = sqrt(5), k>=1, h>=0, are jointly ranked.

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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

A182847 Joint-rank array of the numbers j*(i-1+r), where r=sqrt(3), i>=1, j>=1, by diagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A292958 Rectangular array by antidiagonals: T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = sqrt(5), k>=1, h>=0, are jointly ranked.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A292958 Rectangular array by antidiagonals: T(n,m) = rank of n(r+m) when all the numbers k(r+h), where r = sqrt(5), k>=1, h>=0, are jointly ranked.