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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

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.

Index entries for sequences of the a(a(n)) = 2n family

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

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[3](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]=0; a[1]=2; a[2]=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 A255527 A060132

Adjacent sequences:  A003602 A003603 A003604 * A003606 A003607 A003608

KEYWORD

nonn,nice

AUTHOR

James Propp

STATUS

approved

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Last modified March 25 01:30 EDT 2017. Contains 284036 sequences.