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.

Showing 1-3 of 3 results.

A379414 a(n) = n + floor(n*s/r) + floor(n*t/r), where r = 3^(1/4), s = 3^(1/2), t = 3^(3/4).

Original entry on oeis.org

3, 7, 11, 15, 19, 23, 28, 31, 35, 40, 44, 47, 52, 56, 59, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 105, 108, 112, 117, 120, 124, 129, 133, 136, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, 181, 185, 189, 194, 197, 201, 206, 210, 213, 218, 222, 225, 230
Offset: 1

Views

Author

Clark Kimberling, Jan 18 2025

Keywords

Comments

This sequence and A379415 and A379416 partition the positive integers; see A184812 for a proof.
For each k in A000027, write "a" if k=A379414(n) for some n, "b" if k=A379415(n) for some n, and "c" if k=A379416(n) for some n. Concatenating these letters for k = 1,2,3,... spells the following infinite word:
cbacbcabccabcbacbcacbcabcbcacbacbcabccbacbcabcacbcbacbcacbacbcbacbcacbcabcbaccbacbcabccabcbacbcacbcabcbcacbacbcabccbacbacbcabccbacbcabcacbcba

Crossrefs

Cf. A184812, A379415, A379416.
Cf. also A011002, A002194, A011022.

Programs

  • Mathematica
    r = 3^(1/4); s = 3^(1/2); t = 3^(3/4);
Table[n + Floor[n*s/r] + Floor[n*t/r], {n, 1, 120}]  (* A379411 *)
Table[n + Floor[n*r/s] + Floor[n*t/s], {n, 1, 120}]  (* A379412 *)
Table[n + Floor[n*r/t] + Floor[n*s/t], {n, 1, 120}]  (* A379413 *)

Formula

a(n) = n + floor(n*r) + floor(n*r^2), where r = 3^(1/4).

A379415 a(n) = n + floor(n*r/s) + floor(n*t/s), where r = 3^(1/4), s = 3^(1/2), t = 3^(3/4).

Original entry on oeis.org

2, 5, 8, 12, 14, 17, 21, 24, 26, 30, 33, 36, 39, 42, 45, 49, 51, 54, 58, 61, 63, 66, 70, 73, 75, 79, 82, 85, 89, 91, 94, 98, 101, 103, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 138, 140, 143, 147, 150, 152, 156, 159, 162, 166, 168, 171, 175, 178, 180
Offset: 1

Views

Author

Clark Kimberling, Jan 18 2025

Keywords

Comments

This sequence and A379414 and A379416 partition the positive integers; see A184812 for a proof.

Crossrefs

Cf. A184812, A379415, A379416.
Cf. also A011002, A002194, A011022.

Programs

  • Mathematica
    r = 3^(1/4); s = 3^(1/2); t = 3^(3/4);
Table[n + Floor[n*s/r] + Floor[n*t/r], {n, 1, 120}]  (* A379414 *)
Table[n + Floor[n*r/s] + Floor[n*t/s], {n, 1, 120}]  (* A379415 *)
Table[n + Floor[n*r/t] + Floor[n*s/t], {n, 1, 120}]  (* A379416 *)

Formula

a(n) = n + floor(n/r) + floor(n*r), where r = 3^(1/4).

A358186 Decimal expansion of the positive real root r of 3*x^4 - 1.

Original entry on oeis.org

7, 5, 9, 8, 3, 5, 6, 8, 5, 6, 5, 1, 5, 9, 2, 5, 4, 7, 3, 3, 1, 1, 8, 7, 7, 5, 0, 6, 5, 4, 5, 4, 5, 3, 3, 5, 3, 9, 6, 7, 7, 3, 4, 4, 8, 8, 8, 7, 3, 1, 0, 3, 1, 8, 6, 1, 1, 2, 8, 8, 6, 5, 3, 7, 0, 0, 3, 2, 2, 2, 3, 3, 1, 7, 1, 0
Offset: 0

Views

Author

Wolfdieter Lang, Dec 04 2022

Keywords

Comments

The other roots are -r and the complex conjugate pair r*i and -r*i.

Examples

			0.75983568565159254733118775065454533539677344888731031861128865370032223317...

Crossrefs

Cf. A011002, A011022.

Programs

  • Mathematica
    RealDigits[Surd[1/3, 4], 10, 120][[1]] (* Amiram Eldar, Dec 07 2022 *)

Formula

r = (1/3)^(1/4) = (1/3)*27^(1/4).
Equals A011022/3.
Equals exp(-arctanh(1/2)/2). - Amiram Eldar, Jul 06 2023
Equals 1/A011002. - Jason Yuen, Jul 27 2024
Showing 1-3 of 3 results.

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