A182846 -id:A182846 - 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}.

A182849 Joint-rank array of the numbers j*(i-1+r), where r = golden ratio = (1+sqrt(5))/2, and i>=1, j>=1, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 6, 7, 4, 9, 13, 11, 5, 14, 19, 21, 16, 8, 18, 27, 31, 30, 22, 10, 23, 36, 43, 45, 39, 28, 12, 26, 44, 56, 61, 57, 50, 34, 15, 32, 52, 68, 78, 79, 73, 60, 40, 17, 37, 63, 83, 94, 101, 98, 87, 70, 47, 20, 42, 72, 96, 113, 124, 126, 118, 104, 82, 54, 24, 48

Offset: 1

Clark Kimberling, Dec 08 2010

Joint-rank arrays are defined in the first comment at A182801. Every positive integer occurs exactly once, so that as a sequence, A182849 is a permutation of the positive integers.

			Northwest corner:
1....3....6....9...
2....7...13...19...
4...11...21...31...
5...16...30...45...

Cf. A182801, A182846.

r=GoldenRatio;
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}]] (* A182849 *)

T(i,j)=SUM{floor(j*(i-1+r)/(k-1+r)): r=(1+sqrt(5))/2, k>=1} for i>=1, j>=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.

A182848 Joint-rank array of the numbers j*(i-1+r), where r=sqrt(5), i>=1, j>=1, by antidiagonals.

Original entry on oeis.org

1, 4, 2, 8, 7, 3, 12, 14, 11, 5, 17, 22, 21, 16, 6, 24, 30, 33, 29, 20, 9, 28, 40, 45, 44, 38, 26, 10, 35, 51, 59, 61, 57, 47, 32, 13, 41, 60, 73, 80, 77, 69, 56, 37, 15, 49, 71, 88, 97, 101, 94, 84, 66, 43, 18, 55, 82, 103, 115, 124, 123, 113, 99, 76, 50, 19, 64

Offset: 1

Clark Kimberling, Dec 08 2010

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

			Northwest corner:
1....4....8...12...17...
2....7...14...22...30...
3...11...21...33...45...
5...16...29...44...61...

Cf. A182801, A182846.

r=Sqrt[5];
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(5).

A292956 Rectangular array by antidiagonals: T(n,m) = rank of n(r+m) when all the numbers k(r+h), where r = sqrt(2), k>=1, h>=0, are jointly ranked.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 04 2017

This is the transpose of the array at A182846. Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers. [Sequence reference corrected by Peter Munn, Aug 27 2022]

			Northwest corner:
1   2    4    6    8    10   14   15   18
3   7    11   17   23   29   36   41   49
5   13   21   30   39   50   62   72   84
9   19   32   46   59   75   90   106  124
12  26   44   61   81   100  120  142  165
The numbers k*(r+h), approximately:
(for k=1):   1.414   2.414   3.414 ...
(for k=2):   2.828   4.828   6.828 ...
(for k=3):   4.242   7.242   10.242 ...
Replacing each by its rank gives
1   2    4
3   7    11
5   13   21

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

Cf. A182846.

r = Sqrt[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]  (* A292956 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292956 sequence *)

T(n,m) = Sum_{k=1...[n + m*n/r]} [1 - r + n*(r + m)/k], where r=sqrt(2) 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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A182849 Joint-rank array of the numbers j*(i-1+r), where r = golden ratio = (1+sqrt(5))/2, and i>=1, j>=1, 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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A182848 Joint-rank array of the numbers j*(i-1+r), where r=sqrt(5), i>=1, j>=1, by antidiagonals.

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A292956 Rectangular array by antidiagonals: T(n,m) = rank of n*(r+m) when all the numbers k*(r+h), where r = sqrt(2), k>=1, h>=0, are jointly ranked.

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Mathematica

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A292956 Rectangular array by antidiagonals: T(n,m) = rank of n(r+m) when all the numbers k(r+h), where r = sqrt(2), k>=1, h>=0, are jointly ranked.