A194959 -id:A194959 - cp's OEIS frontend

A194981 Interspersion fractally induced by A194979, a rectangular array, by antidiagonals.

1, 2, 3, 4, 6, 5, 7, 10, 8, 9, 11, 15, 12, 14, 13, 16, 21, 17, 20, 18, 19, 22, 28, 23, 27, 24, 25, 26, 29, 36, 30, 35, 31, 32, 34, 33, 37, 45, 38, 44, 39, 40, 43, 41, 42, 46, 55, 47, 54, 48, 49, 53, 50, 52, 51, 56, 66, 57, 65, 58, 59, 64, 60, 63, 61, 62, 67, 78, 68

Offset: 1

Clark Kimberling, Sep 07 2011

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. Every pair of rows eventually intersperse. As a sequence, A194981 is a permutation of the positive integers, with inverse A194982.

			Northwest corner:
1...2...4...7...11..16..22
3...6...10..15..21..28..36
5...8...12..17..23..30..38
9...14..20..27..35..44..54
13..18..24..31..39..48..58

Cf. A194959, A019480, A194982.

r = Sqrt[3]; p[n_] := 1 + Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A194979 = 1+ A097337 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A194980 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]]  (* A194981 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194982 *)

A194974 Interspersion fractally induced by A194973, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 12, 16, 19, 20, 21, 17, 18, 22, 26, 27, 28, 23, 24, 25, 29, 34, 35, 36, 30, 31, 32, 33, 37, 43, 44, 45, 38, 40, 41, 42, 39, 46, 53, 54, 55, 47, 50, 51, 52, 48, 49, 56, 64, 65, 66, 57, 61, 62, 63, 58, 59, 60, 67, 76, 77

Offset: 1

Clark Kimberling, Sep 07 2011

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence.

			Northwest corner:
1...2...4...7...11
3...5...8...13..19
6...9...14..20..27
10..15..21..28..36
12..17..23..30..38

Cf. A194959, A194973, A194975.

p[n_] := Floor[(n + 3)/4] + Mod[n - 1, 4]
Table[p[n], {n, 1, 90}]  (* A053737(n+4), n>=0 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20]  (* A194973 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]]  (* A194974 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194975 *)

A194980 Fractalization of (1+[n/sqrt(3)]), where [ ]=floor.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 3, 4, 2, 1, 3, 5, 4, 2, 1, 3, 5, 6, 4, 2, 1, 3, 5, 6, 7, 4, 2, 1, 3, 5, 6, 8, 7, 4, 2, 1, 3, 5, 6, 8, 9, 7, 4, 2, 1, 3, 5, 6, 8, 10, 9, 7, 4, 2, 1, 3, 5, 6, 8, 10, 11, 9, 7, 4, 2, 1, 3, 5, 6, 8, 10, 12, 11, 9, 7, 4, 2, 1, 3, 5, 6, 8, 10, 12, 13, 11, 9, 7, 4, 2, 1, 3, 5

Offset: 1

Clark Kimberling, Sep 07 2011

nonn

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. The sequence (1+[n/sqrt(3)]) is A194979.

Cf. A194959, A194879, A194981, A194982.

r = Sqrt[3]; p[n_] := 1 + Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A194979 = 1+ A097337 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A194980 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]]  (* A194981 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194982 *)

A194988 Interspersion fractally induced by A194987, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 10, 7, 9, 8, 15, 11, 14, 12, 13, 21, 16, 20, 17, 19, 18, 28, 22, 27, 23, 26, 25, 24, 36, 29, 35, 30, 34, 33, 31, 32, 45, 37, 44, 38, 43, 42, 39, 41, 40, 55, 46, 54, 47, 53, 52, 48, 51, 49, 50, 66, 56, 65, 57, 64, 63, 58, 62, 59, 61, 60, 78, 67, 77

Offset: 1

Clark Kimberling, Sep 07 2011

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. Every pair of rows eventually intersperse. As a sequence, A194988 is a permutation of the positive integers, with inverse A194989.

			Northwest corner:
1...3...6...10..15..21
2...4...7...11..16..22
5...9...14..20..27..35
8...12..17..23..30..38
13..19..26..34..43..53

Index entries for sequences that are permutations of the natural numbers

Cf. A194959, A194986, A194988, A194989.

r = Sqrt[6]; p[n_] := 1 + Floor[n/r]
Table[p[n], {n, 1, 90}] (* A194986 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A194987 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]] (* A194988 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194989 *)

A064578 Inverse permutation to A057027.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 7, 9, 10, 8, 11, 13, 15, 14, 12, 16, 18, 20, 21, 19, 17, 22, 24, 26, 28, 27, 25, 23, 29, 31, 33, 35, 36, 34, 32, 30, 37, 39, 41, 43, 45, 44, 42, 40, 38, 46, 48, 50, 52, 54, 55, 53, 51, 49, 47, 56, 58, 60, 62, 64, 66, 65, 63, 61, 59, 57, 67, 69, 71, 73, 75, 77

Offset: 1

N. J. A. Sloane, Oct 16 2001

The sequence is an intra-block permutation of positive integers. - Boris Putievskiy, Mar 13 2024

			From _Boris Putievskiy_, Mar 13 2024: (Start)
Start of the sequence as a triangular array T(n,k) read by rows:
       k=1   2   3   4   5   6
  n=1:   1;
  n=2:   2,  3;
  n=3:   4,  6,  5;
  n=4:   7,  9, 10,  8;
  n=5:  11, 13, 15, 14, 12;
  n=6:  16, 18, 20, 21, 19, 17;
Row n contains a permutation block of the n numbers (n-1)*n/2+1, (n-1)*n/2+2, ..., (n-1)*n/2+n to themselves.
Subtracting (n-1)*n/2 from each term in row n gives A194959, in which each row is a permutation of 1..n:
  1;
  1, 2;
  1, 3, 2;
  1, 3, 4, 2;
  1, 3, 5, 4, 2;
  1, 3, 5, 6, 4, 2; (End)

Boris Putievskiy, Table of n, a(n) for n = 1..9870
Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Index entries for sequences that are permutations of the natural numbers

Cf. A057027, A057944, A194959.

T[n_, k_] := (n - 1)*n/2 + Min[2*k - 1, 2*(n - k + 1)];
Nmax = 6; Table[T[n, k], {n, 1, Nmax}, {k, 1, n}] // Flatten (* Boris Putievskiy, Mar 29 2024 *)

a(n) = my(A = (sqrtint(8*n) + 1)\2, B = A*(A - 1)/2, C = n - B); B + if(C <= (A+1)\2, 2*C - 1, 2*(A - C + 1)) \\ Mikhail Kurkov, Mar 12 2024

From Boris Putievskiy, Mar 29 2024: (Start)
a(n) = A057944(n-1) + A194959(n).
T(n,k) = (n-1)*n/2 + min(2*k-1, 2*(n-k+1)), for 1 <= k <= n.
(End)

More terms from Vladeta Jovovic, Oct 18 2001

A194914 Fractalization of (1+[n/sqrt(8)]), where [ ]=floor.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 4, 3, 1, 2, 5, 4, 3, 1, 2, 5, 6, 4, 3, 1, 2, 5, 7, 6, 4, 3, 1, 2, 5, 8, 7, 6, 4, 3, 1, 2, 5, 8, 9, 7, 6, 4, 3, 1, 2, 5, 8, 10, 9, 7, 6, 4, 3, 1, 2, 5, 8, 11, 10, 9, 7, 6, 4, 3, 1, 2, 5, 8, 11, 12, 10, 9, 7, 6, 4, 3, 1, 2, 5, 8, 11, 13, 12, 10, 9, 7, 6, 4, 3, 1, 2, 5, 8

Offset: 1

Clark Kimberling, Sep 08 2011

nonn

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. The sequence (1+[n/sqrt(8)]) is A194990.

Cf. A194959, A194914, A194915, A194916.

r = Sqrt[8]; p[n_] := 1 + Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A194990  *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A194914 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]]  (* A194915 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194916 *)

A194961 Fractalization of A194960.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 4, 2, 3, 1, 4, 5, 2, 3, 1, 4, 5, 6, 2, 3, 1, 4, 7, 5, 6, 2, 3, 1, 4, 7, 8, 5, 6, 2, 3, 1, 4, 7, 8, 9, 5, 6, 2, 3, 1, 4, 7, 10, 8, 9, 5, 6, 2, 3, 1, 4, 7, 10, 11, 8, 9, 5, 6, 2, 3, 1, 4, 7, 10, 11, 12, 8, 9, 5, 6, 2, 3, 1, 4, 7, 10, 13, 11, 12, 8, 9, 5, 6, 2, 3, 1, 4, 7

Offset: 1

Clark Kimberling, Sep 06 2011

nonn

See A194959 for a discussion of fractalization.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A194959, A194960, A194962, A194963.

p[n_] := Floor[(n + 2)/3] + Mod[n - 1, 3]
Table[p[n], {n, 1, 90}]  (* A194960 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20]  (* A194961 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13}, {k, 1, n}]]  (* A194962 *)
q[n_] := Position[w, n]; Flatten[
Table[q[n], {n, 1, 80}]]  (* A194963 *)

A194962 Interspersion fractally induced by A194960.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 8, 11, 14, 15, 12, 13, 16, 20, 21, 17, 18, 19, 22, 27, 28, 23, 25, 26, 24, 29, 35, 36, 30, 33, 34, 31, 32, 37, 44, 45, 38, 42, 43, 39, 40, 41, 46, 54, 55, 47, 52, 53, 48, 50, 51, 49, 56, 65, 66, 57, 63, 64, 58, 61, 62, 59, 60, 67, 77, 78, 68, 75, 76, 69, 73, 74, 70, 71, 72

Offset: 1

Clark Kimberling, Sep 07 2011

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence.

			Northwest corner:
   1...2...4...7..11..16..22
   3...5...9..14..20..27..35
   6..10..15..21..28..36..45
   8..12..17..23..30..38..47
  18..13..25..33..42..52..63
Antidiagonals of the array:
   1;
   2,  3;
   4,  5,  6;
   7,  9, 10,  8;
  11, 14, 15, 12, 13;
  16, 20, 21, 17, 18, 19;
  22, 27, 28, 23, 25, 26, 24;
  29, 35, 36, 30, 33, 34, 31, 32;
  37, 44, 45, 38, 42, 43, 39, 40, 41;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A194959, A194960, A194962, A194963.

p[n_] := Floor[(n + 2)/3] + Mod[n - 1, 3]
Table[p[n], {n, 1, 90}]  (* A194960 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20]  (* A194961 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13}, {k, 1, n}]]  (* A194962 *)
q[n_] := Position[w, n]; Flatten[
Table[q[n], {n, 1, 80}]]  (* A194963 *)

A194963 Inverse permutation of A194962; every positive integer occurs exactly once.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 8, 9, 11, 14, 15, 12, 13, 16, 19, 20, 21, 17, 18, 22, 25, 28, 26, 27, 23, 24, 29, 32, 35, 36, 33, 34, 30, 31, 37, 40, 43, 44, 45, 41, 42, 38, 39, 46, 49, 52, 55, 53, 54, 50, 51, 47, 48, 56, 59, 62, 65, 66, 63, 64, 60, 61, 57, 58, 67, 70, 73

Offset: 1

Clark Kimberling, Sep 07 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A194959, A194960, A194961, A194962.

Mathematica
```
(See A194962.)
```

A194977 Interspersion fractally induced by A194976, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 9, 11, 12, 15, 13, 14, 16, 17, 21, 18, 19, 20, 22, 23, 28, 24, 25, 27, 26, 29, 30, 36, 31, 32, 35, 33, 34, 37, 38, 45, 39, 40, 44, 41, 42, 43, 46, 47, 55, 48, 49, 54, 50, 51, 52, 53, 56, 57, 66, 58, 59, 65, 60, 61, 62, 64, 63, 67, 68, 78, 69, 70, 77, 71, 72, 73, 76, 74, 75

Offset: 1

Clark Kimberling, Sep 07 2011

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. Every pair of rows eventually intersperse. As a sequence, A194977 is a permutation of the positive integers, with inverse A194978.

			Northwest corner:
   1  2  4  7 11 16 22
   3  5  8 12 17 23 30
   6 10 15 21 28 36 45
   9 13 18 24 31 39 48
  14 19 25 32 40 49 59

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A194959, A194976, A194978.

r = Sqrt[2]; p[n_] := 1 + Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A049474 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A194976 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]] (* A194977 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]] (* A194978 *)

Terms a(70) and beyond from G. C. Greubel, Mar 28 2018

cp's OEIS Frontend

A194981 Interspersion fractally induced by A194979, a rectangular array, by antidiagonals.

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A194974 Interspersion fractally induced by A194973, a rectangular array, by antidiagonals.

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A194980 Fractalization of (1+[n/sqrt(3)]), where [ ]=floor.

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A194988 Interspersion fractally induced by A194987, a rectangular array, by antidiagonals.

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A064578 Inverse permutation to A057027.

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A194914 Fractalization of (1+[n/sqrt(8)]), where [ ]=floor.

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A194961 Fractalization of A194960.

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A194962 Interspersion fractally induced by A194960.

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A194963 Inverse permutation of A194962; every positive integer occurs exactly once.

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