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 1-5 of 5 results.

A182760 Beatty sequence for (3 + 5^(-1/2))/2.

Original entry on oeis.org

1, 3, 5, 6, 8, 10, 12, 13, 15, 17, 18, 20, 22, 24, 25, 27, 29, 31, 32, 34, 36, 37, 39, 41, 43, 44, 46, 48, 49, 51, 53, 55, 56, 58, 60, 62, 63, 65, 67, 68, 70, 72, 74, 75, 77, 79, 81, 82, 84, 86, 87, 89, 91, 93, 94, 96, 98, 99, 101, 103, 105, 106, 108, 110, 112, 113, 115, 117, 118, 120, 122, 124, 125, 127, 129, 130, 132, 134, 136, 137, 139, 141, 143, 144, 146, 148, 149, 151, 153, 155, 156, 158, 160, 162, 163, 165, 167, 168
Offset: 1

Views

Author

Clark Kimberling, Nov 28 2010

Keywords

Comments

Suppose that u and v are positive real numbers for which the sets S(u)={j*u} and S(v)={k*v}, for j>=1 and k>=1, are disjoint. Let a(n) be the position of n*u when the numbers in S(u) and S(v) are jointly ranked. Then, as is easy to prove, a is the Beatty sequence of the number r=1+u/v, and the complement of a is the Beatty sequence of s=1+v/u. For A182760, take u = golden ratio = (1+sqrt(5))/2 and v=sqrt(5), so that r=(3+5^(-1/2))/2 and s=(7-sqrt(5))/2.

Examples

			Let u=(1+sqrt(5))/2 and v=sqrt(5).  When the numbers ju and kv are jointly ranked, we write U for numbers of the form ju and V for the others.  Then the ordering of the ranked numbers is given by U V U V U U V U V U V U U ..  The positions of U are given by A182760.

Links

Crossrefs

Cf. A182761 (the complement of A182760), A242671

Programs

Formula

a(n) = floor(r*n), where r = (3 + 5^(-1/2))/2 = 1.72360...

Extensions

More than the usual number of terms are shown in order to distinguish this from a very similar sequence. - N. J. A. Sloane, Jan 20 2025

A285677 {0010->2}-transform of the infinite Fibonacci word A003849.

Original entry on oeis.org

0, 1, 2, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 2
Offset: 1

Views

Author

Clark Kimberling, May 11 2017

Keywords

Comments

As a word, A003849 = 01001010010010100..., and replacing each 0010 by 2 gives 0121201012120101201012120101212010...
Warning: "replacing each 0010 by 2" means "replacing each 0010 by 2 from left to right, consecutively". The result is that the word a(8)...a(14)=0010010 in A003849 is replaced by 201, not by 22. - Michel Dekking, Aug 27 2018

Links

Crossrefs

Cf. A003849, A284620, A285678, A182761, A285679.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 13] ; (* A003849 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"0010" -> "2"}]
st = ToCharacterCode[w1] - 48; (* A285677 *)
Flatten[Position[st, 0]];  (* A285678 *)
Flatten[Position[st, 1]];  (* A182761 - conjectured *)
Flatten[Position[st, 2]];  (* A285679 *)

A285679 Positions of 2 in A285677.

Original entry on oeis.org

3, 5, 10, 12, 17, 22, 24, 29, 31, 36, 41, 43, 48, 53, 55, 60, 62, 67, 72, 74, 79, 81, 86, 91, 93, 98, 103, 105, 110, 112, 117, 122, 124, 129, 134, 136, 141, 143, 148, 153, 155, 160, 162, 167, 172, 174, 179, 184, 186, 191, 193, 198, 203, 205, 210, 212, 217
Offset: 1

Views

Author

Clark Kimberling, May 11 2017

Keywords

Comments

A 3-way partition of the positive integers, by positions of 0, 1, 2 in A285677:
A285678: positions of 0; slope t = (4+sqrt(5))/2;
A182761: positions of 1; slope u = (7-sqrt(5))/2;
A285679: positions of 2; slope v = (1+3*sqrt(5))/2;
where 1/t + 1/u + 1/v = 1.
Conjecture: a(n) - a(n-1) is in {2,5} for n>=2.
See A285683 for a proof of this conjecture. - Michel Dekking, Oct 09 2018
a(n) = A285683(n-1) for n>1, see A285683 for a proof. - Michel Dekking, Oct 09 2018

Links

Crossrefs

Cf. A003849, A284620, A285678, A182761, A285679.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 13] ; (* A003849 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"0010" -> "2"}]
st = ToCharacterCode[w1] - 48; (* A285677 *)
Flatten[Position[st, 0]];  (* A285678 *)
Flatten[Position[st, 1]];  (* A182761 *)
Flatten[Position[st, 2]];  (* A285679 *)

Formula

a(n) = 3*floor((n-1)*phi) - n + 4

A190004 A190002/2.

Original entry on oeis.org

2, 4, 7, 9, 11, 14, 16, 19, 21, 23, 26, 28, 30, 33, 35, 38, 40, 42, 45, 47, 50, 52, 54, 57, 59, 61, 64, 66, 69, 71, 73, 76, 78, 80, 83, 85, 88, 90, 92, 95, 97, 100, 102, 104, 107, 109, 111, 114, 116, 119, 121, 123, 126, 128, 130, 133, 135, 138, 140, 142, 145, 147, 150, 152, 154, 157, 159, 161, 164, 166, 169, 171, 173, 176
Offset: 1

Views

Author

Clark Kimberling, May 03 2011

Keywords

Comments

See A180002.
First differs from A182761 at n=55: a(55)=130, A182761(55)=131. - Bruno Berselli, Jun 04 2013

Links

Crossrefs

Cf. A190002, A190003.

Programs

  • Magma
    [(n + Floor(n*(Sinh(1))^2) + Floor(n*(Cosh(1))^2))/2: n in [1..100]]; // G. C. Greubel, Jan 11 2018
  • Mathematica
    r=1; s=Sinh[1]^2; t=Cosh[1]^2;
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
Table[a[n], {n, 1, 120}]  (* A190002 *)
Table[b[n], {n, 1, 120}]  (* A190003 *)
Table[c[n], {n, 1, 120}]  (* A005408 *)
Table[a[n]/2, {n, 1, 120}](* A190004 *)
Table[b[n]/2, {n, 1, 120}](* A182760 *)
  • PARI
    for(n=1,100, print1((n + floor(n*(sinh(1))^2) + floor(n*(cosh(1))^2))/2, ", ")) \\ G. C. Greubel, Jan 11 2018

Formula

A190002: a(n) = n + [n*(sinh(1))^2] + [n*(cosh(1))^2].
A190003: b(n) = n + [n*(csch(1))^2] + [n*(coth(1))^2].
A190004: a(n)/2 = (n + [n*(sinh(1))^2] + [n*(cosh(1))^2])/2.
A005408: c(n) = 2*n - 1.

A285678 Positions of 0 in A285677.

Original entry on oeis.org

1, 6, 8, 13, 15, 18, 20, 25, 27, 32, 34, 37, 39, 44, 46, 49, 51, 56, 58, 63, 65, 68, 70, 75, 77, 82, 84, 87, 89, 94, 96, 99, 101, 106, 108, 113, 115, 118, 120, 125, 127, 130, 132, 137, 139, 144, 146, 149, 151, 156, 158, 163, 165, 168, 170, 175, 177, 180, 182
Offset: 1

Views

Author

Clark Kimberling, May 11 2017

Keywords

Comments

A 3-way partition of the positive integers, by positions of 0, 1, 2 in A285677:
A285678: positions of 0; slope t = (4+sqrt(5))/2;
A182761: positions of 1; slope u = (7-sqrt(5))/2;
A285679: positions of 2; slope v = (1+3*sqrt(5))/2;
where 1/t + 1/u + 1/v = 1. Conjecture: a(n) - a(n-1) is in {2,3,4,5} for n>=2.

Links

Crossrefs

Cf. A003849, A284620, A285678, A182761, A285679.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 13] ; (* A003849 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"0010" -> "2"}]
st = ToCharacterCode[w1] - 48; (* A285677 *)
Flatten[Position[st, 0]];  (* A285678 *)
Flatten[Position[st, 1]];  (* A182761 *)
Flatten[Position[st, 2]];  (* A285679 *)
Showing 1-5 of 5 results.

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

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