cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A194104 Natural interspersion of A194102; a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 7, 4, 5, 12, 8, 9, 6, 19, 13, 14, 10, 11, 27, 20, 21, 15, 16, 17, 36, 28, 29, 22, 23, 24, 18, 47, 37, 38, 30, 31, 32, 25, 26, 59, 48, 49, 39, 40, 41, 33, 34, 35, 73, 60, 61, 50, 51, 52, 42, 43, 44, 45, 88, 74, 75, 62, 63, 64, 53, 54, 55, 56, 46, 104, 89, 90
Offset: 1

Views

Author

Clark Kimberling, Aug 15 2011

Keywords

Comments

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194100 is a permutation of the positive integers; its inverse is A194101.

Examples

			Northwest corner:
1...3...7...12...19
2...4...8...13...20
5...9...14..21...29
6...10..15..22...30
11..16..23..31...40

Crossrefs

Cf. A194029, A194102, A194103, A194105.

Programs

  • Mathematica
    z = 40; g = Sqrt[2];
c[k_] := Sum[Floor[j*g], {j, 1, k}];
c = Table[c[k], {k, 1, z}]  (* A194102 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 800}]  (* A194103  new *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 16}, {k, 1, n}]]  (* A194104 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194105 *)

A361989 a(n) is the sum of the Fibonacci numbers missing from the dual Zeckendorf representation of n; a(0) = 0, and for n > 0, a(n) = A022290(A035327(A003754(n+1))).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 0, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0
Offset: 0

Views

Author

Rémy Sigrist, Apr 02 2023

Keywords

Comments

We consider that a Fibonacci number is missing from the dual Zeckendorf representation of a number if it does not appear in this representation and a larger Fibonacci number appears in it.
The dual Zeckendorf representation is also known as the lazy Fibonacci representation (see A356771 for further details).
This sequence can also be seen as an irregular table T(n, k), n > 0, k = 1..A000045(n), where T(n, k) = A000045(n) - k.
a(n-1) for n>=1 is the starting position of the first occurrence of one of the longest words w in the Fibonacci word A003849 such that no length-n factor of w is repeated. The length of such words is 2n. (See links) - Gandhar Joshi, Mar 19 2024

Examples

			For n = 42:
- using F(k) = A000045(k),
- the dual Zeckendorf representation of 42 is F(8) + F(7) + F(5) + F(3) + F(2),
- the numbers F(6) and F(4) are missing,
- so a(42) = F(6) + F(4) = 8 + 3 = 11.
.
As an irregular triangle the sequence begins:
     0;
     0;
     1,  0;
     2,  1,  0;
     4,  3,  2, 1, 0;
     7,  6,  5, 4, 3, 2, 1, 0;
    12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0;
    ...

Links

Crossrefs

Cf. A000045, A003754, A022290, A035327, A072649, A130312, A132665, A194029, A356771.

Programs

  • PARI
    for (n = 1, 9, for (k = 1, f = fibonacci(n), print1 (f-k", ")))

Formula

a(n) = A000045(A072649(n)) - A194029(n) for n > 0.
a(n) = A130312(n) - A194029(n) for n > 0.

A194011 Natural interspersion of A002061; a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 7, 4, 5, 13, 8, 9, 6, 21, 14, 15, 10, 11, 31, 22, 23, 16, 17, 12, 43, 32, 33, 24, 25, 18, 19, 57, 44, 45, 34, 35, 26, 27, 20, 73, 58, 59, 46, 47, 36, 37, 28, 29, 91, 74, 75, 60, 61, 48, 49, 38, 39, 30, 111, 92, 93, 76, 77, 62, 63, 50, 51, 40, 41, 133, 112
Offset: 1

Views

Author

Clark Kimberling, Aug 15 2011

Keywords

Comments

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194011 is a permutation of the positive integers; its inverse is A194012.

Examples

			Northwest corner:
1...3...7...13...21...31
2...4...8...14...22...32
5...9...15..23...33...45
6...10..16..24...34...46
11..17..25..35...47...61

Crossrefs

Cf. A194029, A002061, A074294, A194012.

Programs

  • Mathematica
    z = 40;
c[k_] := k^2 - k + 1
c = Table[c[k], {k, 1, z}]  (* A002061 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 800}]  (* A074294 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 16}, {k, 1, n}]]  (* A194011 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194012 *)

A194032 Natural interspersion of the squares (1,4,9,16,25,...), a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 4, 2, 9, 5, 3, 16, 10, 6, 7, 25, 17, 11, 12, 8, 36, 26, 18, 19, 13, 14, 49, 37, 27, 28, 20, 21, 15, 64, 50, 38, 39, 29, 30, 22, 23, 81, 65, 51, 52, 40, 41, 31, 32, 24, 100, 82, 66, 67, 53, 54, 42, 43, 33, 34, 121, 101, 83, 84, 68, 69, 55, 56, 44, 45
Offset: 1

Views

Author

Clark Kimberling, Aug 12 2011

Keywords

Comments

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194032 is a permutation of the positive integers; its inverse is A194033.

Examples

			Northwest corner:
  1...4...9...16...25
  2...5...10..17...26
  3...6...11..18...27
  7...12..19..28...39
  8...13..20..29...40

Links

Crossrefs

Cf. A000290, A071797, A194033, A192872.

Programs

  • Mathematica
    z = 30;
c[k_] := k^2;
c = Table[c[k], {k, 1, z}]  (* A000290 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]] (* A071797 *)
f = Table[f[n], {n, 1, 255}]
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 14}, {k, 1, n}]] (* A194032 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]] (* A194033 *)

Formula

T(n, k) = (k + max(floor(n/2)-1,0))^2 + n - 1. - Zhuorui He, Jul 08 2025

A194034 Natural interspersion of A028387, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 5, 2, 11, 6, 3, 19, 12, 7, 4, 29, 20, 13, 8, 9, 41, 30, 21, 14, 15, 10, 55, 42, 31, 22, 23, 16, 17, 71, 56, 43, 32, 33, 24, 25, 18, 89, 72, 57, 44, 45, 34, 35, 26, 27, 109, 90, 73, 58, 59, 46, 47, 36, 37, 28, 131, 110, 91, 74, 75, 60, 61, 48, 49, 38, 39, 155, 132
Offset: 1

Views

Author

Clark Kimberling, Aug 12 2011

Keywords

Comments

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194034 is a permutation of the positive integers; its inverse is A194035.

Examples

			Northwest corner:
1...5...11...19...29...41
2...6...12...20...30...42
3...7...13...21...31...43
4...8...14...22...32...44
9...15..23...33...45...59

Crossrefs

Cf. A194029, A194035.

Programs

  • Mathematica
    z = 30;
c[k_] := k^2 + k - 1;
c = Table[c[k], {k, 1, z}]  (* A028387 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 255}]  (* A074294 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 13}, {k, 1, n}]]  (* A194034 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]]  (* A194035 *)

A194036 Natural interspersion of A028872, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 6, 2, 13, 7, 3, 22, 14, 8, 4, 33, 23, 15, 9, 5, 46, 34, 24, 16, 10, 11, 61, 47, 35, 25, 17, 18, 12, 78, 62, 48, 36, 26, 27, 19, 20, 97, 79, 63, 49, 37, 38, 28, 29, 21, 118, 98, 80, 64, 50, 51, 39, 40, 30, 31, 141, 119, 99, 81, 65, 66, 52, 53, 41, 42, 32, 166, 142
Offset: 1

Views

Author

Clark Kimberling, Aug 12 2011

Keywords

Comments

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194036 is a permutation of the positive integers; its inverse is A194037.

Examples

			Northwest corner:
1...6...13...22...33
2...7...14...23...34
3...8...15...24...35
4...9...16...25...36
5...10..17...26...37
11..18..27...38...51

Crossrefs

Cf. A192872, A194037.

Programs

  • Mathematica
    z = 30;
c[k_] := k^2 + 2 k - 2;
c = Table[c[k], {k, 1, z}]  (* A028872 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 255}]  (* A071797 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 13}, {k, 1, n}]]  (* A194036 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]]  (* A194037 *)

A194038 Natural interspersion of A034856, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 4, 2, 8, 5, 3, 13, 9, 6, 7, 19, 14, 10, 11, 12, 26, 20, 15, 16, 17, 18, 34, 27, 21, 22, 23, 24, 25, 43, 35, 28, 29, 30, 31, 32, 33, 53, 44, 36, 37, 38, 39, 40, 41, 42, 64, 54, 45, 46, 47, 48, 49, 50, 51, 52, 76, 65, 55, 56, 57, 58, 59, 60, 61, 62, 63, 89, 77, 66
Offset: 0

Views

Author

Clark Kimberling, Aug 12 2011

Keywords

Comments

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194038 is a permutation of the positive integers; its inverse is A194040.

Examples

			Northwest corner:
1...4...8...13...19
2...5...9...14...20
3...6...10..15...21
7...11..16..22...29
12..17..23..30...38

Crossrefs

Cf. A192872, A194040.

Programs

  • Mathematica
    z = 30;
c[k_] := (k^2 + 3 k - 2)/2;
c = Table[c[k], {k, 1, z}]  (* A034856 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 255}]  (* essentially A002260 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 13}, {k, 1, n}]]   (* A194038 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]]  (* A194040 *)

A194051 Natural interspersion of A194050, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 6, 4, 9, 10, 7, 8, 16, 17, 11, 12, 13, 27, 28, 18, 19, 20, 14, 45, 46, 29, 30, 31, 21, 15, 74, 75, 47, 48, 49, 32, 22, 23, 121, 122, 76, 77, 78, 50, 33, 34, 24, 197, 198, 123, 124, 125, 79, 51, 52, 35, 25, 320, 321, 199, 200, 201, 126, 80, 81, 53
Offset: 1

Views

Author

Clark Kimberling, Aug 13 2011

Keywords

Comments

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194051 is a permutation of the positive integers; its inverse is A194052.

Examples

			Northwest corner:
1...2...5...9...16
3...6...10..17..28
4...7...11..18..29
8...12..19..30..48
13..20..31..49..78

Crossrefs

Cf. A194029, A014739, A194050, A104052.

Programs

  • Mathematica
    z = 50;
c[k_] := LucasL[k + 1] - 2;
c = Table[c[k], {k, 1, z}]  (* A014739 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 600}]  (* A194050 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]] (* A194051 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 90}]] (* A194052 *)

A194054 Natural interspersion of A054347; a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 4, 2, 8, 5, 3, 14, 9, 6, 7, 22, 15, 10, 11, 12, 31, 23, 16, 17, 18, 13, 42, 32, 24, 25, 26, 19, 20, 54, 43, 33, 34, 35, 27, 28, 21, 68, 55, 44, 45, 46, 36, 37, 29, 30, 84, 69, 56, 57, 58, 47, 48, 38, 39, 40, 101, 85, 70, 71, 72, 59, 60, 49, 50, 51, 41, 120, 102
Offset: 1

Views

Author

Clark Kimberling, Aug 15 2011

Keywords

Comments

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194054 is a permutation of the positive integers; its inverse is A194055.

Examples

			Northwest corner:
1...4...8...14...22...31
2...5...9...15...23...32
3...6...10..16...24...33
7...11..17..25...34...45

Crossrefs

Cf. A194029, A054347, A194053, A194058.

Programs

  • Mathematica
    z = 40; g = GoldenRatio
c[k_] := Sum[Floor[j*g], {j, 1, k}];
c = Table[c[k], {k, 1, z}]  (* A054347 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 800}]  (* A194053 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 16}, {k, 1, n}]]  (* A194054 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194058 *)

A194056 Natural interspersion of A000071(Fibonacci numbers minus 1), a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 11, 20, 21, 22, 23, 16, 17, 33, 34, 35, 36, 24, 25, 18, 54, 55, 56, 57, 37, 38, 26, 19, 88, 89, 90, 91, 58, 59, 39, 27, 28, 143, 144, 145, 146, 92, 93, 60, 40, 41, 29, 232, 233, 234, 235, 147, 148, 94, 61, 62, 42, 30
Offset: 1

Views

Author

Clark Kimberling, Aug 13 2011

Keywords

Comments

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194056 is a permutation of the positive integers; its inverse is A194057.

Examples

			Northwest corner:
1...2...4...7...12
3...5...8...13..21
6...9...14..22..35
10..15..23..36..57
11..16..24..37..58

Crossrefs

Cf. A194029, A194055, A000071, A000045, A194057.

Programs

  • Mathematica
    z = 50;
c[k_] := -1 + Fibonacci[k + 2]
c = Table[c[k], {k, 1, z}] (* A000071, F(n+2)-1 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 300}]   (* A194055 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 11}, {k, 1, n}]] (* A194056 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]]  (* A194057 *)
Previous Showing 11-20 of 39 results. Next

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