A285677 -id:A285677

Search: a285677 -id:a285677

Displaying 1-3 of 3 results found.

page 1

A285679

Positions of 2 in A285677.

+20
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

Clark Kimberling, Table of n, a(n) for n = 1..10000

FORMULA

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

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 *)

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 11 2017

STATUS

approved

A285678

Positions of 0 in A285677.

+20
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

Clark Kimberling, Table of n, a(n) for n = 1..10000

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 *)

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 11 2017

STATUS

approved

A285680

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..10000

EXAMPLE

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

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 *)

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 11 2017

STATUS

approved

page 1

Search completed in 0.006 seconds