A182801 -id:A182801 - cp's OEIS frontend

A283939 Interspersion of the signature sequence of sqrt(2).

1, 3, 2, 6, 5, 4, 11, 9, 8, 7, 17, 15, 13, 12, 10, 25, 22, 20, 18, 16, 14, 34, 31, 28, 26, 23, 21, 19, 44, 41, 38, 35, 32, 29, 27, 24, 56, 52, 49, 46, 42, 39, 36, 33, 30, 69, 65, 61, 58, 54, 50, 47, 43, 40, 37, 84, 79, 75, 71, 67, 63, 59, 55, 51, 48, 45, 100

Offset: 1

Clark Kimberling, Mar 19 2017

Row n is the ordered sequence of numbers k such that A007336(k)=n. As a sequence, A283939 is a permutation of the positive integers. As an array, A283939 is the joint-rank array (defined at A182801) of the numbers {i+j*r}, for i>=1, j>=1, where r = sqrt(2). This is a transposable interspersion; i.e., every row intersperses all other rows, and every column intersperses all other columns.

			Northwest corner:
  1   3   6   11   17   25   34   44   56
  2   5   9   15   22   31   41   52   65
  4   8   13  20   28   38   49   61   75
  7   12  18  26   35   46   58   71   86
  10  16  23  32   42   54   67   81   97
  14  21  29  39   50   63   77   91   109

Clark Kimberling, Antidiagonals n = 1..60, flattened
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Cf. A007336, A002193, A022776, A283962.

r = Sqrt[2]; z = 100;
s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r];
u = Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}] (* A022776, col 1 of A283939 *)
v = Table[s[n], {n, 0, z}] (* A022775, row 1 of A283939*)
w[i_, j_] := u[[i]] + v[[j]] + (i - 1)*(j - 1) - 1;
Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283939, array *)
p = Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283939, sequence *)

r = sqrt(2);
z = 100;
s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r));
p(n) = n + 1 + sum(k=0, n, floor((n - k)/r));
u = v = vector(z + 1);
for(n=1, 101, (v[n] = s(n - 1)));
for(n=1, 101, (u[n] = p(n - 1)));
w(i, j) = u[i] + v[j] + (i - 1) * (j - 1) - 1;
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(k, n - k + 1), ", "); );print(); ); };
tabl(10) \\ Indranil Ghosh, Mar 21 2017

sqrt2 = 2 ** 0.5
def s(n): return 1 if n<1 else s(n - 1) + 1 + int(n*sqrt2)
def p(n): return n + 1 + sum([int((n - k)/sqrt2) for k in range(0, n+1)])
v=[s(n) for n in range(0, 101)]
u=[p(n) for n in range(0, 101)]
def w(i,j): return u[i - 1] + v[j - 1] + (i - 1) * (j - 1) - 1
for n in range(1, 11):
    print ([w(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 21 2017

A292963 Rectangular array by antidiagonals: T(n,m) = rank of n(e + m) when all the numbers k(e+h), for k>=1, h>=0, are jointly ranked.

Original entry on oeis.org

1, 2, 4, 3, 7, 9, 5, 11, 15, 14, 6, 16, 22, 24, 20, 8, 19, 29, 34, 32, 27, 10, 25, 38, 45, 48, 43, 35, 12, 30, 46, 57, 62, 61, 54, 42, 13, 36, 55, 70, 79, 81, 76, 67, 50, 17, 40, 64, 83, 95, 101, 100, 92, 78, 58, 18, 47, 73, 97, 113, 122, 125, 120, 108, 89

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
4    7    11   16   19   25
9    15   22   29   38   46
14   24   34   45   57   70
20   32   48   62   79   95
27   43   61   81   101  122
35   54   76   100  125  152
42   67   92   120  151  181
The numbers k*(r+h), approximately:
(for k=1):   2.718  3.718  4.718 ...
(for k=2):   5.436   7.436   9.436 ...
(for k=3):   8.154   11.854   14.154 ...
Replacing each by its rank gives
1    2    3
4    7    14
9    15   22

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

Cf. A182801, A292964.

r = E; 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]  (* A292963 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292963 sequence *)

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

A292964 Rectangular array by antidiagonals: T(n,m) = rank of n(1/e + m) when all the numbers k(1/e+h), for k>=1, h>=0, are jointly ranked.

Original entry on oeis.org

1, 4, 2, 8, 10, 3, 13, 19, 16, 5, 17, 29, 32, 23, 6, 22, 40, 48, 44, 30, 7, 27, 52, 65, 68, 58, 37, 9, 34, 63, 82, 93, 89, 72, 46, 11, 38, 76, 102, 118, 120, 108, 87, 53, 12, 43, 88, 123, 144, 153, 149, 132, 101, 60, 14, 50, 99, 141, 171, 187, 189, 178, 155

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      4     8      13     17     22
2     10     19     29     40     52
3     16     32     48     65     82
5     23     44     68     93     118
6     30     58     89     120    153
7     37     72     108    149    189
9     46     87     132    178    228
The numbers k*(1/e+h), approximately:
(for k=1):   0.367   1.367  2.3667 ...
(for k=2):   0.735   2.735  4.735 ...
(for k=3):   1.103   4.103  7.103 ...
Replacing each by its rank gives
1    4     8
2    10    19
3    16    32

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

Cf. A182801, A292963.

r = 1/E; 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]  (* A292964 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292964 sequence *)

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

A157927 Joint-rank array of the numbers i^2+j^2, where i>=0, j>=0.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 7, 5, 5, 7, 10, 8, 6, 8, 10, 14, 11, 9, 9, 11, 14, 19, 15, 13, 12, 13, 15, 19, 24, 20, 16, 14, 14, 16, 20, 24, 30, 25, 21, 18, 17, 18, 21, 25, 30, 37, 31, 27, 23, 22, 22, 23, 27, 31, 37, 44, 38, 32, 28, 26, 25, 26, 28, 32, 38, 44

Offset: 1

Clark Kimberling, Dec 17 2010

The definition of joint-rank array given at A182801 is
here extended to arrays R={f(i,j)} for which the numbers
f(i,j) are not necessarily distinct.  Specifically, all
duplicates are assigned the same rank when all the numbers
in R are jointly ranked.  Let {a(i,j)} denote the
resulting joint-rank array.  In case all f(i,j) are
positive integers, a(i,j)=f(i,j)-L(i,j), where L(i,j) is
the number of numbers in R that are <=f(i,j).
(Row 1)=A047808.

			A corner of the array R={i^2+j^2} is
0....1....4....9...16...
1....2....5...10...17...
4....5....8...13...20...
9...10...13...18...25...
Replace each term of R by its rank:
1....2....4....7...10...
2....3....5....8...11...
4....5....6....9...13...
7....8....9...12...14...

Cf. A182801, A048147.

A182833 Joint-rank array of the numbers j*e^(i-1), read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 5, 12, 21, 30, 6, 15, 34, 61, 84, 8, 19, 44, 94, 170, 232, 9, 24, 55, 124, 257, 466, 635, 11, 28, 68, 155, 342, 702, 1272, 1731, 13, 32, 79, 188, 427, 935, 1912, 3465, 4711, 14, 37, 91, 219, 515, 1169, 2548, 5201, 9426, 12814, 16, 41, 103, 251

Offset: 1

Clark Kimberling, Dec 07 2010

Joint-rank arrays are defined in the first comment at A182801. (Column 1)=A117869. Every positive integer occurs exactly once, so that as a sequence, A182833 is a permutation of the positive integers.

			Northwest corner:
1....2....4....5....6...
3....7...12...15...19...
10..21...34...44...55...
30..61...94..124..155...

Cf. A182801.

T(i,j)=SUM{floor(j*e^(i-n)): n>=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.

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.

A292962 Rectangular array by antidiagonals: T(n,m) = rank of n(r-1+m) when all the numbers k(r+h), where r = log(2), k>=1, h>=0, are jointly ranked.

Original entry on oeis.org

1, 3, 2, 5, 7, 4, 9, 14, 13, 6, 11, 21, 24, 19, 8, 16, 29, 36, 35, 26, 10, 18, 38, 50, 53, 46, 32, 12, 23, 45, 63, 72, 68, 59, 41, 15, 27, 56, 77, 90, 94, 87, 73, 47, 17, 30, 65, 92, 110, 119, 117, 106, 84, 54, 20, 34, 74, 107, 132, 146, 150, 142, 125, 98

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    5    9    11   16   18
2    7    14   21   29   38   45
4    13   24   36   50   63   77
6    19   35   53   72   90   110
8    26   46   68   94   119  146
10   32   59   87   117  150  181
12   41   73   106  142  180  219
The numbers k*(r+h), approximately:
(for k=1):   0.693   1.693   2.693 ...
(for k=2):   1.386   3.386   5.386 ...
(for k=3):   2.079   5.079   8.079 ...
Replacing each by its rank gives
1    3    5
2    7    14
4    13   24

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

Cf. A182801.

r = Log[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]  (* A292962 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292962 sequence *)

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

A292965 Rectangular array by antidiagonals: T(n,m) = rank of n(Pi + m) when all the numbers k(Pi+h), for k >= 1, h >= 0, are jointly ranked.

Original entry on oeis.org

1, 2, 5, 3, 8, 10, 4, 12, 16, 17, 6, 15, 22, 26, 23, 7, 20, 30, 35, 36, 31, 9, 25, 38, 46, 50, 47, 39, 11, 29, 45, 58, 64, 65, 59, 48, 13, 34, 54, 70, 78, 84, 79, 71, 56, 14, 41, 63, 83, 95, 103, 104, 97, 86, 67, 18, 44, 73, 94, 113, 123, 127, 124, 115, 99

Offset: 1

Clark Kimberling, Oct 06 2017

Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.

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

Cf. A182801.

r = Pi; 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]  (* A292965 array *)
Table[t[n - k + 1, k - 1], {n, 1, z}, {k, n, 1, -1}] // Flatten  (* A292965 sequence *)

T(n,m) = Sum_{k=1...[n + m*n/Pi]} [1 - Pi + n*(Pi + m)/k], where [ ]=floor.
Northwest corner:
1      2     3     4     6     7
5      8     12    15    20    25
10     16    22    30    38    45
17     26    35    46    58    70
23     36    50    64    78    95
31     47    65    84    103   123
39     59    79    104   127   153
The numbers k*(Pi+h), approximately:
(for k=1):   3.141   4.141   5.141 ...
(for k=2):   6.283   8.283   10.283 ...
(for k=3):   9.424   12.424  15.424 ...
Replacing each by its rank gives
1    2    3
5    8    12
10   16   22

cp's OEIS Frontend

A283939 Interspersion of the signature sequence of sqrt(2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

A292963 Rectangular array by antidiagonals: T(n,m) = rank of n*(e + m) when all the numbers k*(e+h), for k>=1, h>=0, are jointly ranked.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A292964 Rectangular array by antidiagonals: T(n,m) = rank of n*(1/e + m) when all the numbers k*(1/e+h), for k>=1, h>=0, are jointly ranked.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A157927 Joint-rank array of the numbers i^2+j^2, where i>=0, j>=0.

Views

Author

Keywords

Comments

Examples

Crossrefs

A182833 Joint-rank array of the numbers j*e^(i-1), read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

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

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

A292962 Rectangular array by antidiagonals: T(n,m) = rank of n*(r-1+m) when all the numbers k*(r+h), where r = log(2), k>=1, h>=0, are jointly ranked.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A292963 Rectangular array by antidiagonals: T(n,m) = rank of n(e + m) when all the numbers k(e+h), for k>=1, h>=0, are jointly ranked.

A292964 Rectangular array by antidiagonals: T(n,m) = rank of n(1/e + m) when all the numbers k(1/e+h), for k>=1, h>=0, are jointly ranked.

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.

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.

A292962 Rectangular array by antidiagonals: T(n,m) = rank of n(r-1+m) when all the numbers k(r+h), where r = log(2), k>=1, h>=0, are jointly ranked.

A292965 Rectangular array by antidiagonals: T(n,m) = rank of n(Pi + m) when all the numbers k(Pi+h), for k >= 1, h >= 0, are jointly ranked.