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.

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

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 *)

A285680 {1010->2}-transform of the infinite Fibonacci word A003849.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 11 2017

Keywords

Examples

			As a word, A003849 = 01001010010010100..., and replacing each 1010 by 2 gives 01002010020201002010020201002020100201...

Links

Crossrefs

Cf. A003849, A285677, A285681, A285682, A285683.

Programs

  • Mathematica
    s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 13] ; (* A003849 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"1010" -> "2"}]
st = ToCharacterCode[w1] - 48; (* A285680 *)
Flatten[Position[st, 0]];  (* A285681 *)
Flatten[Position[st, 1]];  (* A285682 *)
Flatten[Position[st, 2]];  (* A285683 *)
Showing 1-3 of 3 results.

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