A194959 -id:A194959 - cp's OEIS frontend

A194983 Fractalization of (1+[n/sqrt(5)]), where [ ]=floor.

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

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(5)]) is A194964.

Cf. A194959, A194983, A194984, A194985.

r = Sqrt[5]; p[n_] := 1 + Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A194964 *)
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] (* A194983 *)
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}]]  (* A194984 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194985 *)

A194984 Interspersion fractally induced by A194964, 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, 24, 25, 36, 29, 35, 30, 34, 31, 33, 32, 45, 37, 44, 38, 43, 39, 42, 40, 41, 55, 46, 54, 47, 53, 48, 52, 49, 51, 50, 66, 56, 65, 57, 64, 58, 63, 59, 62, 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, A194984 is a permutation of the positive integers, with inverse A194985.

			Northwest corner:
1...3...6...10..15..21..28
2...4...7...11..16..22..29
5...9...14..20..27..35..44
8...12..17..23..30..38..47
13..19..26..34..43..53..64

Cf. A194959, A194964, A194983, A194985.

r = Sqrt[5]; p[n_] := 1 + Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A194964 *)
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] (* A194983 *)
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}]]  (* A194984 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194985 *)

A194987 Fractalization of (1+[n/sqrt(6)]), where [ ]=floor.

Original entry on oeis.org

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

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(6)]) is A194986.

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 *)

A194989 Inverse permutation of A194988; every positive integer occurs exactly once.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 07 2011

nonn

Index entries for sequences that are permutations of the natural numbers

Cf. A194988, A194987, A194959.

Mathematica
```
(See A194988.)
```

A330081 If the binary expansion of n is (b(1), ..., b(w)), then the binary expansion of a(n) is (b(1), b(3), b(5), ..., b(6), b(4), b(2)).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 10, 12, 14, 9, 11, 13, 15, 16, 20, 18, 22, 24, 28, 26, 30, 17, 21, 19, 23, 25, 29, 27, 31, 32, 36, 40, 44, 34, 38, 42, 46, 48, 52, 56, 60, 50, 54, 58, 62, 33, 37, 41, 45, 35, 39, 43, 47, 49, 53, 57, 61, 51, 55, 59, 63, 64, 72, 68, 76

Offset: 0

Rémy Sigrist, Dec 01 2019

This sequence is a permutation of the nonnegative integers that preserves the binary length as well as the Hamming weight. See A330090 for the inverse.

			For n = 1234:
- the binary expansion of 1234 is "10011010010",
- odd-indexed bits are "101100",
- even-indexed bits are "01001", and in reverse order "10010",
- hence the binary expansion of a(1234) is "10110010010",
- so a(1234) = 1426.

See A329303 for a similar sequence.

Cf. A003558, A194959, A330090 (inverse).

shuffle(v) = { my (w=vector(#v), o=0, e=#v+1); for (k=1, #v, w[if (k%2, o++, e--)]=v[k]); w }
a(n) = fromdigits(shuffle(binary(n)), 2)

If n has w binary digits, then a^A003558(w-1)(n) = n (where a^k denotes the k-th iterate of the sequence).

A194915 Interspersion fractally induced by A194990, a rectangular array, by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 08 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, A194915 is a permutation of the positive integers, with inverse A194916.

			Northwest corner:
1...3...6...10..15..21
2...4...7...11..16..22
5...9...14..20..27..35
8...13..19..26..34..43
12..17..23..30..38..47

Cf. A194959, A194990, 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 *)

A194921 Fractalization of (n - [n/sqrt(2)]), where [ ]=floor.

Original entry on oeis.org

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

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 (n-[n/sqrt(2)]) is A194920.

G. C. Greubel, Table of n, a(n) for n = 1..5050

Cf. A194920, A194922, A195071.

r = Sqrt[2]; p[n_] := n - Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A194920 *)
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]  (* A194921 *)
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}]]  (* A194922 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]   (* A195071 *)

A194922 Interspersion fractally induced by A194920, a rectangular array, by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 08 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, A194922 is a permutation of the positive integers, with inverse A195071.

			Northwest corner:
   1,  3,  6, 10, 15, 21
   2,  5,  9, 14, 20, 27, 35
   4,  7, 11, 16, 22, 29, 37
   8, 13, 19, 26, 34, 43, 53
  12, 18, 25, 33, 42, 52, 63

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

Cf. A194920, A194921, A195071.

r = Sqrt[2]; p[n_] := n - Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A194920 *)
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]  (* A194921 *)
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}]]  (* A194922 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]   (* A195071 *)

A194966 Interspersion fractally induced by A194965, a rectangular array, by antidiagonals.

Original entry on oeis.org

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

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, A194966 is a permutation of the positive integers, with inverse A194967.

			Northwest corner:
1...2...4...7...11..16
3...5...8...12..18..25
6...9...13..19..26..34
10..14..20..27..35..44
15..21..28..36..45..55

Index entries for sequences that are permutations of the natural numbers

Cf. A194959, A194965, A194967 (inverse).

p[n_] := Floor[(n + 4)/5] + Mod[n - 1, 5]
Table[p[n], {n, 1, 90}]  (* A053824(n+5), 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]   (* A194965 *)
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}]]  (* A194966 *)
q[n_] := Position[w, n]; Flatten[
Table[q[n], {n, 1, 80}]]  (* A194967 *)

A194969 Interspersion fractally induced by A194968, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 7, 10, 8, 9, 11, 15, 12, 13, 14, 16, 21, 17, 18, 20, 19, 22, 28, 23, 24, 27, 25, 26, 29, 36, 30, 31, 35, 32, 34, 33, 37, 45, 38, 39, 44, 40, 43, 41, 42, 46, 55, 47, 48, 54, 49, 53, 50, 51, 52, 56, 66, 57, 58, 65, 59, 64, 60, 61, 63, 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, A194969 is a permutation of the positive integers, with inverse A194970.

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

Index entries for sequences that are permutations of the natural numbers

Cf. A194958, A019446, A194968, A194970 (inverse).

r = GoldenRatio; p[n_] := 1 + Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A019446 *)
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]  (* A194968 *)
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}]]  (* A194969 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194970 *)

cp's OEIS Frontend

A194983 Fractalization of (1+[n/sqrt(5)]), where [ ]=floor.

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A194984 Interspersion fractally induced by A194964, a rectangular array, by antidiagonals.

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A194987 Fractalization of (1+[n/sqrt(6)]), where [ ]=floor.

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Mathematica

A194989 Inverse permutation of A194988; every positive integer occurs exactly once.

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Mathematica

A330081 If the binary expansion of n is (b(1), ..., b(w)), then the binary expansion of a(n) is (b(1), b(3), b(5), ..., b(6), b(4), b(2)).

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A194915 Interspersion fractally induced by A194990, a rectangular array, by antidiagonals.

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Mathematica

A194921 Fractalization of (n - [n/sqrt(2)]), where [ ]=floor.

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A194922 Interspersion fractally induced by A194920, a rectangular array, by antidiagonals.

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Mathematica

A194966 Interspersion fractally induced by A194965, a rectangular array, by antidiagonals.

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