A168224 Where record values occur in A168223.
0, 5, 7, 11, 17, 19, 23, 29, 31, 35, 41, 43, 47, 53, 55, 59, 65, 67, 71, 77, 79, 83, 89, 91, 95, 101, 103, 107, 113, 115, 119, 125, 127, 131, 137, 139, 143, 149, 151, 155, 161, 163, 167, 173, 175, 179, 185, 187, 191, 197, 199, 203, 209, 211, 215, 221, 223, 227, 233
Offset: 1
Keywords
A006368 The "amusical permutation" of the nonnegative numbers: a(2n)=3n, a(4n+1)=3n+1, a(4n-1)=3n-1.
0, 1, 3, 2, 6, 4, 9, 5, 12, 7, 15, 8, 18, 10, 21, 11, 24, 13, 27, 14, 30, 16, 33, 17, 36, 19, 39, 20, 42, 22, 45, 23, 48, 25, 51, 26, 54, 28, 57, 29, 60, 31, 63, 32, 66, 34, 69, 35, 72, 37, 75, 38, 78, 40, 81, 41, 84, 43, 87, 44, 90, 46, 93, 47, 96, 49, 99, 50, 102, 52, 105, 53
Offset: 0
Comments
A permutation of the nonnegative integers.
There is a famous open question concerning the closed trajectories under this map - see A217218, A028393, A028394, and Conway (2013).
This is lodumo_3 of A131743. - Philippe Deléham, Oct 24 2011
Multiples of 3 interspersed with numbers other than multiples of 3. - Harvey P. Dale, Dec 16 2011
For n>0: a(2n+1) is the smallest number missing from {a(0),...,a(2n-1)} and a(2n) = a(2n-1) + a(2n+1). - Bob Selcoe, May 24 2017
From Wolfdieter Lang, Sep 21 2021: (Start)
The permutation P of positive natural numbers with P(n) = a(n-1) + 1, for n >= 1, is the inverse of the permutation given in A265667, and it maps the index n of A178414 to the index of A047529: A178414(n) = A047529(P(n)).
Thus each number {1, 3, 7} (mod 8) appears in the first column A178414 of the array A178415 just once. For the formulas see below. (End)
Starting at n = 1, the sequence equals the smallest unused positive number such that a(n)-a(n-1) does not appear as a term in the current sequence. - Scott R. Shannon, Dec 20 2023
Examples
9 is odd so a(9) = round(3*9/4) = round(7-1/4) = 7.
References
- J. H. Conway, Unpredictable iterations, in Proc. Number Theory Conf., Boulder, CO, 1972, pp. 49-52.
- R. K. Guy, Unsolved Problems in Number Theory, E17.
- J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010; see page 5.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- R. Zumkeller, Table of n, a(n) for n = 0..10000
- David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669 [math.NT], 2015 and J. Int. Seq. 18 (2015) 15.6.7..
- J. H. Conway, On unsettleable arithmetical problems, Amer. Math. Monthly, 120 (2013), 192-198. [Introduces the name "amusical permutation".]
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
- S. Schreiber & N. J. A. Sloane, Correspondence, 1980
- Index entries for sequences related to 3x+1 (or Collatz) problem
- Index entries for two-way infinite sequences
- Index entries for sequences that are permutations of the natural numbers
- Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).
Crossrefs
Inverse mapping to A006369.
Programs
-
Haskell
a006368 n | u' == 0 = 3 * u | otherwise = 3 * v + (v' + 1) `div` 2 where (u,u') = divMod n 2; (v,v') = divMod n 4 -- Reinhard Zumkeller, Apr 18 2012 -
Magma
[n mod 2 eq 1 select Round(3*n/4) else 3*n/2: n in [0..80]]; // G. C. Greubel, Jan 03 2024
-
Maple
f:=n-> if n mod 2 = 0 then 3*n/2 elif n mod 4 = 1 then (3*n+1)/4 else (3*n-1)/4; fi; # N. J. A. Sloane, Jan 21 2011 A006368:=(1+3*z+z**2+3*z**3+z**4)/(1+z**2)/(z-1)**2/(1+z)**2; # [Conjectured (correctly, except for the offset) by Simon Plouffe in his 1992 dissertation.]
-
Mathematica
Table[If[EvenQ[n],(3n)/2,Floor[(3n+2)/4]],{n,0,80}] (* or *) LinearRecurrence[ {0,1,0,1,0,-1},{0,1,3,2,6,4},80] (* Harvey P. Dale, Dec 16 2011 *) -
PARI
a(n)=(3*n+n%2)\(2+n%2*2)
-
PARI
a(n)=if(n%2,round(3*n/4),3*n/2)
-
Python
def a(n): return 0 if n == 0 else 3*n//2 if n%2 == 0 else (3*n+1)//4 print([a(n) for n in range(72)]) # Michael S. Branicky, Aug 12 2021
Formula
If n even, then a(n) = 3*n/2, otherwise, a(n) = round(3*n/4).
G.f.: x*(1+3*x+x^2+3*x^3+x^4)/((1-x^2)*(1-x^4)). - Michael Somos, Jul 23 2002
a(n) = -a(-n).
From Reinhard Zumkeller, Nov 20 2009: (Start)
A168221(n) = a(a(n)).
a(n) = a(n-2) + a(n-4) - a(n-6); a(0)=0, a(1)=1, a(2)=3, a(3)=2, a(4)=6, a(5)=4. - Harvey P. Dale, Dec 16 2011
From Wolfdieter Lang, Sep 21 2021: (Start)
Formulas for the permutation P(n) = a(n-1) + 1 mentioned above:
P(n) = n + floor(n/2) if n is odd, and n - floor(n/4) if n is even.
P(n) = (3*n-1)/2 if n is odd; P(n) = (3*n+2)/4 if n == 2 (mod 4); and P(n) = 3*n/4 if n == 0 (mod 4). (End)
Extensions
Edited by Michael Somos, Jul 23 2002
I replaced the definition with the original definition of Conway and Guy. - N. J. A. Sloane, Oct 03 2012
A006369 a(n) = 2*n/3 for n divisible by 3, otherwise a(n) = round(4*n/3). Or, equivalently, a(3*n-2) = 4*n-3, a(3*n-1) = 4*n-1, a(3*n) = 2*n.
0, 1, 3, 2, 5, 7, 4, 9, 11, 6, 13, 15, 8, 17, 19, 10, 21, 23, 12, 25, 27, 14, 29, 31, 16, 33, 35, 18, 37, 39, 20, 41, 43, 22, 45, 47, 24, 49, 51, 26, 53, 55, 28, 57, 59, 30, 61, 63, 32, 65, 67, 34, 69, 71, 36, 73, 75, 38, 77, 79, 40, 81, 83, 42, 85, 87, 44, 89, 91, 46, 93, 95
Offset: 0
Comments
Original name was: Nearest integer to 4n/3 unless that is an integer, when 2n/3.
This function was studied by Lothar Collatz in 1932.
Fibonacci numbers lodumo_2. - Philippe Deléham, Apr 26 2009
a(n) = A006368(n) + A168223(n); A168222(n) = a(a(n)); A168221(a(n)) = A006368(n). - Reinhard Zumkeller, Nov 20 2009
The permutation P given in A265667 is P(n) = a(n-1) + 1, for n >= 0, with a(-1) = -1. Observed by Kevin Ryde. - Wolfdieter Lang, Sep 22 2021
Examples
G.f. = x + 3*x^2 + 2*x^3 + 5*x^4 + 7*x^5 + 4*x^6 + 9*x^7 + 11*x^8 + 6*x^9 + ...
References
- R. K. Guy, Unsolved Problems in Number Theory, E17.
- M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 579-581.
- K. Knopp, Infinite Sequences and Series, Dover Publications, NY, 1958, p. 77.
- J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010; see page 31 (g(n)) and page 270 (f(n)).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
- J. H. Conway, On unsettleable arithmetical problems, Amer. Math. Monthly, 120 (2013), 192-198.
- M. Klamkin, Proposer, An infinite permutation, Problem 63-13, SIAM Review, Vol. 8:2 (1966), 234-236.
- J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
- S. Schreiber & N. J. A. Sloane, Correspondence, 1980.
- Index entries for two-way infinite sequences
- Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).
- Index entries for sequences related to 3x+1 (or Collatz) problem.
- Index entries for sequences that are permutations of the natural numbers.
Crossrefs
Programs
-
Haskell
a006369 n | m > 0 = round (4 * fromIntegral n / 3) | otherwise = 2 * n' where (n',m) = divMod n 3 -- Reinhard Zumkeller, Dec 31 2011 -
Maple
A006369 := proc(n) if n mod 3 = 0 then 2*n/3 else round(4*n/3); fi; end; f:=proc(N) if N mod 3 = 0 then 2*(N/3); elif N mod 3 = 2 then 4*((N+1)/3)-1; else 4*((N+2)/3)-3; fi; end; # N. J. A. Sloane, Feb 04 2011 A006369:=(1+z**2)*(z**2+3*z+1)/(z-1)**2/(z**2+z+1)**2; # Simon Plouffe, in his 1992 dissertation
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Mathematica
Table[If[Divisible[n,3],(2n)/3,Floor[(4n)/3+1/2]],{n,0,80}] (* Harvey P. Dale, Nov 03 2011 *) Table[n + Floor[(n + 1)/3] (-1)^Mod[n + 1, 3], {n, 0, 80}] (* Bruno Berselli, Dec 10 2015 *) -
PARI
{a(n) = if( n%3, round(4*n / 3), 2*n / 3)}; /* Michael Somos, Oct 05 2003 */
Formula
From Michael Somos, Oct 05 2003: (Start)
G.f.: x * (1 + 3*x + 2*x^2 + 3*x^3 + x^4) / (1 - x^3)^2.
a(3*n) = 2*n, a(3*n + 1) = 4*n + 1, a(3*n - 1) = 4*n - 1, a(n) = -a(-n) for all n in Z. (End)
The map is: n -> if n mod 3 = 0 then 2*n/3 elif n mod 3 = 1 then (4*n-1)/3 else (4*n+1)/3.
a(n) = (2 - ((2*n + 1) mod 3) mod 2) * floor((2*n + 1)/3) + (2*n + 1) mod 3 - 1. - Reinhard Zumkeller, Jan 23 2005
a(n) = lod_2(F(n)). - Philippe Deléham, Apr 26 2009
0 = 21 + a(n)*(18 + 4*a(n) - a(n+1) - 7*a(n+2)) + a(n+1)*(-a(n+2)) + a(n+2)*(-18 + 4*a(n+2)) for all n in Z. - Michael Somos, Aug 24 2014
a(n) = n + floor((n+1)/3)*(-1)^((n+1) mod 3). - Bruno Berselli, Dec 10 2015
a(n) = 2*a(n-3) - a(n-6) for n >= 6. - Werner Schulte, Mar 16 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = log(sqrt(2)+2)/sqrt(2) + (1-sqrt(2)/2)*log(2)/2. - Amiram Eldar, Sep 29 2022
Extensions
New name from Jon E. Schoenfield, Jul 28 2015
A047342 Numbers that are congruent to {0, 3, 4} mod 7.
0, 3, 4, 7, 10, 11, 14, 17, 18, 21, 24, 25, 28, 31, 32, 35, 38, 39, 42, 45, 46, 49, 52, 53, 56, 59, 60, 63, 66, 67, 70, 73, 74, 77, 80, 81, 84, 87, 88, 91, 94, 95, 98, 101, 102, 105, 108, 109, 112, 115, 116, 119, 122, 123, 126, 129, 130, 133, 136, 137, 140
Offset: 1
Comments
Record values in A168223: a(n) = A168223(A168224(n)) and A168223(m) < a(n) for m < A168224(n). - Reinhard Zumkeller, Nov 20 2009
Also: Numbers n such that kronecker(n^2-4,7) = -1. - M. F. Hasler, Mar 14 2013
Links
- Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Programs
-
Magma
[n : n in [0..150] | n mod 7 in [0, 3, 4]]; // Wesley Ivan Hurt, Jun 13 2016
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Maple
A047342:=n->(21*n-21-6*cos(2*n*Pi/3)-2*sqrt(3)*sin(2*n*Pi/3))/9: seq(A047342(n), n=1..100); # Wesley Ivan Hurt, Jun 13 2016
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Mathematica
Select[Range[0,150], MemberQ[{0,3,4}, Mod[#,7]]&] (* Harvey P. Dale, Mar 18 2011 *) CoefficientList[Series[(3x+x^2+3x^3)/((-1+x)^2(1+x+x^2)),{x,0,160}],x] (* Vladimir Joseph Stephan Orlovsky, Jan 26 2012 *) LinearRecurrence[{1, 0, 1, -1},{0, 3, 4, 7},61] (* Ray Chandler, Aug 25 2015 *)
Formula
G.f.: x(3+x+3x^2)/((1-x)^2*(1+x+x^2)). - R. J. Mathar, Sep 17 2008
a(n) = a(n-1) + a(n-3) - a(n-4), n>4. - Vincenzo Librandi, Mar 24 2011
From Wesley Ivan Hurt, Jun 13 2016: (Start)
a(n) = (21*n-21-6*cos(2*n*Pi/3)-2*sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 7k-3, a(3k-1) = 7k-4, a(3k-2) = 7k-7. (End)
Comments
Links
Programs
Mathematica
With[{nn=300},DeleteDuplicates[Thread[{Range[0,nn-1],LinearRecurrence[{-2,-2,0,3,4,3,0,-2,-2,-1},{0,0,0,0,-1,3,-5,4,-1,-1},nn]}],GreaterEqual[#1[[2]],#2[[2]]]&]][[;;,1]] (* Harvey P. Dale, Mar 01 2024 *)