This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A003605 Unique monotonic sequence of nonnegative integers satisfying a(a(n)) = 3n. (Formerly M0747) 11
 0, 2, 3, 6, 7, 8, 9, 12, 15, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Another definition: a(0) = 0, a(1) = 2; for n > 1, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a multiple of 3". - Benoit Cloitre, Feb 14 2003 Yet another definition: a(0) = 0, a(1)=2; for n > 1, a(n) is the smallest integer > a(n-1) satisfying "if n is in the sequence, a(n)==0 (mod 3)" ("only if" omitted). This sequence is the case m = 2 of the following family: a(1, m) = m, a(n, m) is the smallest integer > a(n-1, m) satisfying "if n is in the sequence, a(n, m) == 0 (mod (2m-1))". The general formula is: for any k >= 0, for j = -m*(2m-1)^k, ..., -1, 0, 1, ..., m*(2m-1)^k, a((m-1)*(2*m-1)^k+j) = (2*m-1)^(k+1)+m*j+(m-1)*abs(j). Numbers whose base 3 representation starts with 2 or ends with 0. - Franklin T. Adams-Watters, Jan 17 2006 REFERENCES Hsien-Kuei Hwang, S Janson, TH Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 J.-P. Allouche, N. Rampersad and J. Shallit, On integer sequences whose first iterates are linear, Aequationes Math. 69 (2005), 114-127. J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29. British Mathematical Olympiad 1992, Problem 5 B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2. (math.NT/0305308) J. Shallit, Number theory and formal languages, in D. A. Hejhal, J. Friedman, M. C. Gutzwiller and A. M. Odlyzko, eds., Emerging Applications of Number Theory, IMA Volumes in Mathematics and Its Applications, V. 109, Springer-Verlag, 1999, pp. 547-570. FORMULA For any k>=0, a(3^k - j) = 2*3^k - 3j, 0 <= j <= 3^(k-1); a(3^k + j) = 2*3^k + j, 0 <= j <= 3^k. a(3n)=3a(n), a(3n+1)=2a(n)+a(n+1), a(3n+2)=a(n)+2a(n+1), n>0. Also a(n+1)-2*a(n)+a(n-1)= { 2 if n=3^k, -2 if n=2*3^k, otherwise 0}, n>1. - Michael Somos, May 03 2000. a(n) = n + A006166(n). - Vladeta Jovovic, Mar 01 2003 a(n) = abs(2*3^floor(log_3(n)) - n) + 2n - 3^floor(log_3(n)) for n>=1. - Theodore Lamort de Gail, Sep 12 2017 EXAMPLE 9 is in the sequence and the smallest multiple of 3 greater than a(9-1)=a(8)=15 is 18. Hence a(9)=18. MAPLE filter:= n ->  (n mod 3 = 0) or (n >= 2*3^floor(log(n))): select(filter, [\$0..1000]); # Robert Israel, Oct 15 2014 MATHEMATICA a[n_] := a[n] = Which[ Mod[n, 3] == 0, 3 a[n/3], Mod[n, 3] == 1, 2*a[(n-1)/3] + a[(n-1)/3 + 1], True, a[(n-2)/3] + 2*a[(n-2)/3 + 1]]; a=0; a=2; a=3; Table[a[n], {n, 0, 67}] (* Jean-François Alcover, Jul 18 2012, after Michael Somos *) PROG (PARI) a(n)=if(n<3, n+(n>0), (3-(n%3))*a(n\3)+(n%3)*a(n\3+1)) (PARI) {A(n)=local(d, w, l3=log(3), l2=log(2), l3n);            l3n = log(n)/l3;            w   = floor(l3n);         \\ highest exponent w such that 3^w <= n            d   = frac(l3n)*l3/l2+1;  \\ first digit in base-3 repr. of n               if ( d<2 , d=1 , d=2 ); \\   make d an integer either 1 or 2            if(d==1, n = n + 3^w , n = (n - 3^w)*3);            return(n); } \\ Gottfried Helms, Jan 11 2012 CROSSREFS Cf. A079000, A007378, A080588, A079351. Sequence in context: A161824 A102806 A275884 * A132188 A326027 A255527 Adjacent sequences:  A003602 A003603 A003604 * A003606 A003607 A003608 KEYWORD nonn,nice AUTHOR STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 22 05:47 EDT 2019. Contains 327287 sequences. (Running on oeis4.)