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A182761 -id:A182761 - OEIS
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Search: a182761 -id:a182761
Displaying 1-5 of 5 results found. page 1
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A182760 Beatty sequence for (3 + 5^(-1/2))/2. +10
19
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Suppose that u and v are positive real numbers for which the sets S(u)={ju} and S(v)={kv}, for j>=1 and k>=1, are disjoint. Let a(n) be the position of nu 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=(7-5^(-1/2)/2.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = floor(r*n), where r = (3 + 5^(-1/2))/2 = 1.72360...
EXAMPLE
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.
MATHEMATICA
Table[Floor[Sqrt[n/20]+3*n/2], {n, 1, 100}] (* G. C. Greubel, Jan 11 2018 *)
PROG
(Magma) [Floor(n*(3+5^(-1/2))/2): n in [1..70]]; // Vincenzo Librandi, Oct 25 2011
(PARI) a(n)=floor(sqrt(n/20)+3*n/2) \\ Charles R Greathouse IV, Jul 02 2013
CROSSREFS
Cf. A182761 (the complement of A182760), A242671
KEYWORD
nonn
AUTHOR
Clark Kimberling, Nov 28 2010
STATUS
approved
A285677 {0010->2}-transform of the infinite Fibonacci word A003849. +10
4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
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
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 - conjectured *)
Flatten[Position[st, 2]]; (* A285679 *)
CROSSREFS
Cf. A003849, A284620, A285678, A182761, A285679.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, May 11 2017
STATUS
approved
A285679 Positions of 2 in A285677. +10
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
A190004 A190002/2. +10
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
See A180002.
First differs from A182761 at n=55: a(55)=130, A182761(55)=131. - Bruno Berselli, Jun 04 2013
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
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.
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 *)
PROG
(PARI) for(n=1, 100, print1((n + floor(n*(sinh(1))^2) + floor(n*(cosh(1))^2))/2, ", ")) \\ G. C. Greubel, Jan 11 2018
(Magma) [(n + Floor(n*(Sinh(1))^2) + Floor(n*(Cosh(1))^2))/2: n in [1..100]]; // G. C. Greubel, Jan 11 2018
CROSSREFS
Cf. A190002, A190003.
KEYWORD
nonn
AUTHOR
Clark Kimberling, May 03 2011
STATUS
approved
A285678 Positions of 0 in A285677. +10
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
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Last modified August 26 19:22 EDT 2024. Contains 375462 sequences. (Running on oeis4.)