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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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A194061 Natural interspersion of A002620; a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 6, 7, 11, 15, 9, 10, 14, 19, 24, 12, 13, 18, 23, 29, 35, 16, 17, 22, 28, 34, 41, 48, 20, 21, 27, 33, 40, 47, 55, 63, 25, 26, 32, 39, 46, 54, 62, 71, 80, 30, 31, 38, 45, 53, 61, 70, 79, 89, 99, 36, 37, 44, 52, 60, 69, 78, 88, 98, 109, 120
Offset: 1

Views

Author

Clark Kimberling, Aug 14 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, A194061 is a permutation of the positive integers; its inverse is A194062.

Examples

			Northwest corner:
1...2...4...6...9...12
3...5...7...10..13..17
8...11..14..18..22..27
15..19..23..28..33..39
24..29..34..40..46..53

Crossrefs

Cf. A194029, A002620, A122197, A194062.

Programs

  • Mathematica
    z = 50;
c[k_] := Floor[((k + 1)^2)/4];
c = Table[c[k], {k, 1, z}]  (* A002620 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 400}]   (* [A122197] *)
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}]] (* A194061 *)
q[n_] := Position[p, n]; Flatten[
 Table[q[n], {n, 1, 90}]]  (* A194062 *)

A194064 Natural interspersion of A006578; a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 4, 2, 8, 5, 3, 14, 9, 6, 7, 21, 15, 10, 11, 12, 30, 22, 16, 17, 18, 13, 40, 31, 23, 24, 25, 19, 20, 52, 41, 32, 33, 34, 26, 27, 28, 65, 53, 42, 43, 44, 35, 36, 37, 29, 80, 66, 54, 55, 56, 45, 46, 47, 38, 39, 96, 81, 67, 68, 69, 57, 58, 59, 48, 49, 50
Offset: 1

Views

Author

Clark Kimberling, Aug 14 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, A194064 is a permutation of the positive integers; its inverse is A194065.

Examples

			Northwest corner:
1...4...8...14...21...30
2...5...9...15...22...31
3...6...10..16...23...32
7...11..17..24...33...43
12..18..25..34...44...56

Crossrefs

Cf. A194029, A194063, A194065.

Programs

  • Mathematica
    z = 50;
c[k_] := k (k + 1)/2 + Floor[(k^2)/4];
c = Table[c[k], {k, 1, z}]  (* A006578 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 400}]   (* A194063 *)
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}]] (* A194064 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 90}]]  (* A194065 *)

A194067 Natural interspersion of A087483; a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 6, 7, 11, 12, 9, 10, 15, 16, 21, 13, 14, 19, 20, 26, 27, 17, 18, 24, 25, 32, 33, 40, 22, 23, 30, 31, 38, 39, 47, 48, 28, 29, 36, 37, 45, 46, 55, 56, 65, 34, 35, 43, 44, 53, 54, 63, 64, 74, 75, 41, 42, 51, 52, 61, 62, 72, 73, 84, 85, 96, 49, 50, 59
Offset: 1

Views

Author

Clark Kimberling, Aug 14 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, A194067 is a permutation of the positive integers; its inverse is A194068.

Examples

			Northwest corner:
1...2...4...6...9...13
3...5...7...10..14..18
8...11..15..19..24..30
12..16..20..25..31..37
21..26..32..38..45..53

Crossrefs

Cf. A194029, A194066, A194068, A087483.

Programs

  • Mathematica
    z = 70;
c[k_] := 1 + Floor[(1/3) k^2];
c = Table[c[k], {k, 1, z}]  (* A087483 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 300}]   (* A194066 *)
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}]] (* A194067 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 90}]] (* A194068 *)

A194070 Natural fractal sequence of A194069.

Original entry on oeis.org

1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
Offset: 1

Views

Author

Clark Kimberling, Aug 14 2011

Keywords

Comments

See A194029 for definitions of natural fractal sequence and natural interspersion.

Crossrefs

Cf. A194029, A194069, A194071.

Programs

  • Mathematica
    z = 70;
c[k_] := 1 + Floor[(2/3) k^2];
c = Table[c[k], {k, 1, z}]  (* A194069 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 300}]   (* A194070 *)
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}]] (* A194071 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 90}]] (* A194072 *)

A194074 Natural fractal sequence of A194073.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15
Offset: 1

Views

Author

Clark Kimberling, Aug 14 2011

Keywords

Comments

See A194029 for definitions of natural fractal sequence and natural interspersion.

Crossrefs

Cf. A194073, A194075.

Programs

  • Mathematica
    z = 70;
c[k_] := 1 + Floor[(3/4) k^2];
c = Table[c[k], {k, 1, z}]  (* A194073 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 300}]   (* A194074 *)
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}]] (* A194075 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 90}]]  (* A194076 *)

A194077 Natural interspersion of A060432; a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 5, 4, 7, 8, 6, 10, 17, 11, 9, 13, 21, 34, 14, 12, 16, 25, 39, 60, 18, 15, 20, 29, 44, 66, 97, 22, 19, 24, 33, 49, 72, 104, 147, 26, 23, 28, 38, 54, 78, 111, 155, 212, 30, 27, 32, 43, 59, 84, 118, 163, 221, 294, 35, 31, 37, 48, 65, 90, 125, 171, 230, 304
Offset: 1

Views

Author

Clark Kimberling, Aug 14 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, A194077 is a permutation of the positive integers; its inverse is A194078.

Examples

			Northwest corner:
1...3...5...8...11...14
2...4...6...9...12...15
7...10..13..16..20...24
17..21..25..29..33..38
34..39..44..49..54..59

Crossrefs

Cf. A194029, A060432, A194078

Programs

  • Mathematica
    z = 70;
c[k_] := Sum[Floor[1/2 + Sqrt[2 j]], {j, 0, k}];
c = Table[c[k], {k, 1, z}]  (* A060432 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 600}]   (* [A121997] *)
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, 12}, {k, 1, n}]] (* A194077 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 90}]]   (* A194078 *)

A194103 Natural fractal sequence of A194102.

Original entry on oeis.org

1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4
Offset: 1

Views

Author

Clark Kimberling, Aug 15 2011

Keywords

Comments

See A194029 for definitions of natural fractal sequence and natural interspersion.

Crossrefs

Cf. A194029, A194102, A194104, A001951.

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 *)

A194107 Natural fractal sequence of A194106.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17
Offset: 1

Views

Author

Clark Kimberling, Aug 15 2011

Keywords

Comments

See A194029 for definitions of natural fractal sequence and natural interspersion.

Crossrefs

Cf. A194029, A194106, A194108.

Programs

  • Mathematica
    z = 40; g = Sqrt[3];
c[k_] := Sum[Floor[j*g], {j, 1, k}];
c = Table[c[k], {k, 1, z}]  (* A194106 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 800}]  (* A194107 *)
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}]]  (* A194108 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194109 *)

A130853 Runs of 1's of lengths 1, Fibonacci numbers F(1), F(2), F(3), ... (A000045) separated by 0's.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Jani Melik, Jul 21 2007

Keywords

Comments

Might be called a Fibonacci message.

Examples

			Begin with 0. First Fibonacci number F(1)=1, so append 1's to 0 once - 01, append 0 - 010, F(2)=1, append 1's once and 0 - 01010, F(3)=2, we append two 1's and 0 - 01010110, ...
As a triangle:
  0, 1;
  0, 1;
  0, 1, 1;
  0, 1, 1, 1;
  0, 1, 1, 1, 1, 1;
  0, 1, 1, 1, 1, 1, 1, 1, 1;
  0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  ...

Links

Crossrefs

Cf. A000045, A093521, A232896 (the positions of zeros).
Cf. A001622, A003849, A005614, A194029.

Programs

  • Maple
    ts_Finonacci_zap:=proc(n) local i,j,tren,ans; ans := [ 0 ]: for i from 1 to n do tren := combinat[fibonacci](i): for j from 1 to tren do ans:=[ op(ans), 1 ]: od: ans:=[ op(ans), 0 ]: od; RETURN(ans) end: ts_Finonacci_zap(16);
# second Maple program:
T:= n-> [0,1$(<<0|1>, <1|1>>^n)[1,2]][]:
seq(T(n), n=1..10);  # Alois P. Heinz, Dec 11 2024
  • Mathematica
    T[n_] := Join[{0}, Table[1, MatrixPower[{{0, 1}, {1, 1}}, n][[1, 2]]]];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, May 21 2025, after Alois P. Heinz *)
  • PARI
    { n=0; i=0; while(n<22, n++; i++; write("b130853.txt", i, " ", 0); k = fibonacci(n); while(k>0, i++; k--; write("b130853.txt", i, " ", 1))); }; \\ Antti Karttunen, Dec 07 2017

Formula

a(n) = b(n+1) - b(n) where b(n) = round(LambertW((phi^(3/2 + n)*log(phi))/sqrt(5)) / log(phi)), phi = (1 + sqrt(5))/2. - Alan Michael Gómez Calderón, Dec 11 2024

A193042 Natural fractal sequence of A194126.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17
Offset: 1

Views

Author

Clark Kimberling, Aug 15 2011

Keywords

Comments

See A194029 for definitions of natural fractal sequence and natural interspersion.

Crossrefs

Cf. A194029, A194126, A194100.

Programs

  • Mathematica
    z = 40; g = GoldenRatio;
c[k_] := -1 + Sum[Floor[j + j*g], {j, 1, k}];
c = Table[c[k], {k, 1, z}]  (* A194126 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 800}]  (* A193042 *)
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}]]  (* A194100 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194101 *)
Previous Showing 21-30 of 39 results. Next

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