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.

Previous Showing 11-20 of 21 results. Next

A167971 Signature sequence of Phi^3 = 4.2360679774998..., where Phi is the golden ratio 1.6180339887499... .

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 1, 10, 6, 2, 11, 7, 3, 12, 8, 4, 13, 9, 5, 1, 14, 10, 6, 2, 15, 11, 7, 3, 16, 12, 8, 4, 17, 13, 9, 5, 1, 18, 14, 10, 6, 2, 19, 15, 11, 7, 3, 20, 16, 12, 8, 4, 21, 17, 13, 9, 5, 22, 1, 18, 14, 10, 6, 23, 2, 19, 15, 11, 7, 24, 3, 20, 16, 12, 8, 25, 4, 21
Offset: 1

Views

Author

Casey Mongoven, Nov 15 2009

Keywords

References

  • Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.

Links

Crossrefs

Cf. A084531, A084532, A167969.

Programs

  • Mathematica
    terms = 86; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j*GoldenRatio^3, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N] &)[[1 ;; terms]] /. GoldenRatio -> 0; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)

A167972 Signature sequence of Phi^4 = 6.8541019662497..., where Phi is the golden ratio 1.6180339887499... .

Original entry on oeis.org

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

Views

Author

Casey Mongoven, Nov 15 2009

Keywords

References

  • Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.

Links

Crossrefs

Cf. A084531, A084532, A167968.

Programs

  • Mathematica
    terms = 83; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j*GoldenRatio^4, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N] &)[[1 ;; terms]] /. GoldenRatio -> 0; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)

A167973 Signature sequence of Phi^5 = 11.090169943749..., where Phi is the golden ratio 1.6180339887499... .

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 13, 2, 14, 3, 15, 4, 16, 5, 17, 6, 18, 7, 19, 8, 20, 9, 21, 10, 22, 11, 23, 12, 1, 24, 13, 2, 25, 14, 3, 26, 15, 4, 27, 16, 5, 28, 17, 6, 29, 18, 7, 30, 19, 8, 31, 20, 9, 32, 21, 10, 33, 22, 11, 34, 23, 12, 1, 35, 24, 13, 2, 36, 25, 14, 3, 37, 26
Offset: 1

Views

Author

Casey Mongoven, Nov 15 2009

Keywords

References

  • Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.

Links

Crossrefs

Cf. A084531, A084532.

Programs

  • Mathematica
    terms = 80; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j*GoldenRatio^5, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N] &)[[1 ;; terms]] /. GoldenRatio -> 0; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)

A167974 Signature sequence of Phi^6 = 17.944271909999..., where Phi is the golden ratio 1.6180339887499... .

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1, 19, 2, 20, 3, 21, 4, 22, 5, 23, 6, 24, 7, 25, 8, 26, 9, 27, 10, 28, 11, 29, 12, 30, 13, 31, 14, 32, 15, 33, 16, 34, 17, 35, 18, 36, 1, 19, 37, 2, 20, 38, 3, 21, 39, 4, 22, 40, 5, 23, 41, 6, 24, 42, 7, 25, 43, 8, 26, 44, 9
Offset: 1

Views

Author

Casey Mongoven, Nov 15 2009

Keywords

References

  • Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.

Links

Crossrefs

Cf. A084531, A084532.

Programs

  • Mathematica
    terms = 79; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j*GoldenRatio^6, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N] &)[[1 ;; terms]] /. GoldenRatio -> 0; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)

A332502 Rectangular array read by antidiagonals: T(n,k) = floor(n + k*r), where r = golden ratio = (1+sqrt(5))/2.

Original entry on oeis.org

0, 1, 1, 3, 2, 2, 4, 4, 3, 3, 6, 5, 5, 4, 4, 8, 7, 6, 6, 5, 5, 9, 9, 8, 7, 7, 6, 6, 11, 10, 10, 9, 8, 8, 7, 7, 12, 12, 11, 11, 10, 9, 9, 8, 8, 14, 13, 13, 12, 12, 11, 10, 10, 9, 9, 16, 15, 14, 14, 13, 13, 12, 11, 11, 10, 10, 17, 17, 16, 15, 15, 14, 14, 13
Offset: 0

Views

Author

Clark Kimberling, May 08 2020

Keywords

Comments

Every nonnegative integer occurs exactly once in the union of row 0 and the main diagonal.
Column 0: Nonnegative integers, A001477.
Row 0: Lower Wythoff sequence, A000201.
Row 1: A026351.
Row 2: A026355 (and essentially A099267).
Main Diagonal: Upper Wythoff sequence, A001950.
Diagonal (1,4,6,9,...) = A003622;
Diagonal (3,5,8,11,...) = A026274;
Diagonal (1,3,6,8,...) = A026352;
Diagonal (2,4,7,9,...) = A026356.

Examples

			Northwest corner:
  0   1   3   4   6   8    9    11   12   14   16
  1   2   4   5   7   9    10   12   13   15   17
  2   3   5   6   8   10   11   13   14   16   18
  3   4   6   7   9   11   12   14   15   17   19
  4   5   7   8   10  12   13   15   16   18   20
  5   6   8   9   11  13   14   16   17   19   21
As a triangle (antidiagonals):
  0
  1   1
  2   2   3
  3   3   4   4
  4   4   5   5   6
  5   5   6   6   7   8
  6   6   7   7   8   9   9

Crossrefs

Cf. A000210, A001950, A001622, A084531, A167267, A019446.

Programs

  • Mathematica
    t[n_, k_] := Floor[n + k*GoldenRatio];
Grid[Table[t[n, k], {n, 0, 10}, {k, 0, 10}]] (* array *)
u = Table[t[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten  (* sequence *)

Formula

T(n,k) = floor(n + k*r), where r = golden ratio = (1+sqrt(5))/2.

A167286 Signature sequence of the smallest Pisot number (A060006).

Original entry on oeis.org

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

Views

Author

Roger L. Bagula, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A007337, A084531

Programs

  • Mathematica
    m = x /. Solve[x^3 - x - 1 == 0, x][[1]]
Take[Transpose[Sort[Flatten[Table[{i + j*m, i}, {i, 25}, {j, 17}], 1], #1[[1]] < #2[[1]] &]][[2]], 95]

A167287 Signature sequence of Pisot number 1.3802775690976206... (A086106).

Original entry on oeis.org

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

Views

Author

Roger L. Bagula, Nov 01 2009

Keywords

Links

Crossrefs

Cf. A007337, A084531

Programs

  • Mathematica
    m = x /. Solve[x^4 - x^3 - 1 == 0, x][[4]]
Take[Transpose[Sort[Flatten[Table[{i + j*m, i}, {i, 25}, {j, 17}], 1], #1[[1]] < #2[[1]] &]][[2]], 95]

A167966 Signature sequence of phi^6 = 0.055728090000841..., where phi is the inverse golden ratio A094214.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1
Offset: 1

Views

Author

Casey Mongoven, Nov 15 2009

Keywords

References

  • Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.

Links

Crossrefs

Cf. A084531, A084532, A167974.

Programs

  • Mathematica
    terms = 105; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j/GoldenRatio^6, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N] &)[[1 ;; terms]] /. GoldenRatio -> \[Infinity]; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)

A167967 Signature sequence of phi^5 = 0.090169943749474..., where phi is the inverse golden ratio A094214.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3
Offset: 1

Views

Author

Casey Mongoven, Nov 15 2009

Keywords

References

  • Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.

Links

Crossrefs

Cf. A084531, A084532, A167973.

Programs

  • Mathematica
    terms = 105; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j/GoldenRatio^5, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N] &)[[1 ;; terms]] /. GoldenRatio -> \[Infinity]; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)

A167975 Signature sequence of Phi^7 = 29.034441853749..., where Phi is the golden ratio 1.6180339887499... .

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 1, 31, 2, 32, 3, 33, 4, 34, 5, 35, 6, 36, 7, 37, 8, 38, 9, 39, 10, 40, 11, 41, 12, 42, 13, 43, 14, 44, 15, 45, 16, 46, 17, 47, 18, 48, 19, 49, 20, 50, 21, 51, 22, 52, 23
Offset: 1

Views

Author

Casey Mongoven, Nov 15 2009

Keywords

References

  • Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.

Links

Crossrefs

Cf. A084531, A084532, A167965.

Programs

  • Mathematica
    terms = 75; m = Ceiling[Sqrt[terms]]; s0 = {}; While[s = (Table[i + j*GoldenRatio^7, {i, 1, m}, {j, 1, m}] // Flatten // SortBy[#, N] &)[[1 ;; terms]] /. GoldenRatio -> 0; s != s0, s0 = s; m = 2 m]; s (* Jean-François Alcover, Jan 08 2017 *)
Previous Showing 11-20 of 21 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.