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 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. (Formerly M2245) 37
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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 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. 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 EXAMPLE 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 + ... 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 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 *) PROG (PARI) {a(n) = if( n%3, round(4*n / 3), 2*n / 3)}; /* Michael Somos, Oct 05 2003 */ (Haskell) a006369 n | m > 0 = round (4 * fromIntegral n / 3) | otherwise = 2 * n' where (n', m) = divMod n 3 -- Reinhard Zumkeller, Dec 31 2011 CROSSREFS Inverse mapping to A006368. Cf. A028397, A069196, A265667. Trajectories under A006368 and A006369: A180853, A217218, A185590, A180864, A028393, A028394, A094328, A094329, A028396, A028395, A217729, A182205, A223083-A223088, A185589. Sequence in context: A265903 A345420 A364885 * A097284 A276684 A105353 Adjacent sequences: A006366 A006367 A006368 * A006370 A006371 A006372 KEYWORD nonn,nice,easy AUTHOR N. J. A. Sloane and J. H. Conway EXTENSIONS New name from Jon E. Schoenfield, Jul 28 2015 STATUS approved

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Last modified April 16 23:37 EDT 2024. Contains 371756 sequences. (Running on oeis4.)