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A005836 Numbers n whose base 3 representation contains no 2.
(Formerly M2353)
112
0, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40, 81, 82, 84, 85, 90, 91, 93, 94, 108, 109, 111, 112, 117, 118, 120, 121, 243, 244, 246, 247, 252, 253, 255, 256, 270, 271, 273, 274, 279, 280, 282, 283, 324, 325, 327, 328, 333, 334, 336, 337, 351, 352 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

3 does not divide binomial(2s,s) if and only if s is a member of this sequence, where binomial(2s,s)= A000984(s) are the central binomial coefficients.

This is the "earliest" sequence obtained among nonnegative numbers by forbidding arithmetic subsequences of length 3. - Robert Craigen (craigenr(AT)cc.umanitoba.ca), Jan 29 2001

In the notation of A185256 this is the Stanley Sequence S(0,1). - N. J. A. Sloane, Mar 19 2010

Complement of A074940. - Reinhard Zumkeller, Mar 23 2003

Sums of distinct powers of 3. - Ralf Stephan, Apr 27 2003

Numbers n such that central trinomial coefficient A002426(n) == 1 (mod 3). - Emeric Deutsch and Bruce E. Sagan, Dec 04 2003

A039966(a(n)+1) = 1; A104406(n) = number of terms <= n.

Subsequence of A125292; A125291(a(n)) = 1 for n>1. - Reinhard Zumkeller, Nov 26 2006

Also final value of n-1 written in base 2 and then read in base 3 and with finally the result translated in base 10. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 23 2007

A081603(a(n)) = 0. - Reinhard Zumkeller, Mar 02 2008

a(n) modulo 2 is the Thue-Morse sequence A010060. - Dennis Tseng, Jul 16 2009

Also numbers such that the balanced ternary representation is the same as the base 3 representation. - Alonso del Arte, Feb 25 2011

Fixed point of the morphism: 0-> 01; 1-> 34; 2-> 67; ...; n-> (3n)(3n+1), starting from a(1)=0. - Philippe Deléham, Oct 22 2011

It appears that this sequence lists the values of n which satisfy the condition sum(binomial(n,k)^(2*j), k=0..n) mod 3 <>0, for any j, with offset 0. See Maple code. - Gary Detlefs, Nov 28 2011

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, E10.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1024

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.

J.-P. Allouche, J. Shallit and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math. 292 (2005) 1-15.

Kathrin Kostorz et al., Distributed coupling complexity in a weakly coupled oscillatory network with associative properties, New J. Phys. 15 (2013), #083010; doi:10.1088/1367-2630/15/8/083010.

C. Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., 274 (2004), 147-160. [From N. J. A. Sloane, Jan 31 2012]

J. W. Layman, Some Properties of a Certain Nonaveraging Sequence, J. Integer Sequences, Vol. 2, 1999, #4.

A. M. Odlyzko and R. P. Stanley, Some curious sequences constructed with the greedy algorithm, 1978, remark 1 (PDF, PS, TeX).

P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 228. [?Broken link]

P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 228.

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

Z. Sunic, Tree morphisms, transducers and integer sequences, arXiv:math.CO/0612080

B. Vasic, K. Pedagani and M. Ivkovic, High-rate girth-eight low-density parity-check codes on rectangular integer lattices, IEEE Transactions on Communications, Vol. 52, Issue 8 (2004), 1248-1252.

Eric Weisstein's World of Mathematics, Central Binomial Coefficient

FORMULA

Numbers n such that the coefficient of x^n is > 0 in prod (k>=0, 1+x^(3^k)). - Benoit Cloitre, Jul 29 2003

a(n+1) = sum( b(k)* 3^k ) for k=0..m and n = sum( b(k)* 2^k ).

a(2n+1)=3a(n+1), a(2n+2)=a(2n+1)+1, a(0)=0.

a(n+1)=3*a(floor(n/2))+n-2*floor(n/2). - Ralf Stephan, Apr 27 2003

G.f. x/(1-x) * Sum(k>=0, 3^k*x^2^k/(1+x^2^k)). - Ralf Stephan, Apr 27 2003

a(n) = Sum_{k = 1..n-1} (1 + 3^A007814(k)) / 2. - Philippe Deléham, Jul 09 2005

If the offset were changed to zero, then: a(0)=0, a(n+1)=f(a(n)+1,f(a(n)+1) where f(x,y) = if x<3 and x<>2 then y else if x mod 3 = 2 then f(y+1, y+1) else f(floor(x/3),y). - Reinhard Zumkeller, Mar 02 2008

With offset a(0)=0: a(n)=Sum_k>=0 {A030308(n,k)*3^k}. - Philippe Deléham, Oct 15 2011

EXAMPLE

a(6) = 12 because 6 = 0*2^0 +1*2^1 +1*2^2 = 2+4 and 12 = 0*3^0 +1*3^1 +1*3^2 = 3+9.

MAPLE

t:=(j, n)-> sum(binomial(n, k)^j, k=0..n):for i from 1 to 400 do if(t(4, i) mod 3 <>0) then print(i) fi od; # Gary Detlefs, Nov 28 2011

MATHEMATICA

Table[FromDigits[IntegerDigits[k, 2], 3], {k, 60}]

Select[Range[0, 400], DigitCount[#, 3, 2]==0&] (* Harvey P. Dale, Jan 04 2012 *)

Join[{0}, Accumulate[Table[(3^IntegerExponent[n, 2]+1)/2, {n, 57}]]] (* IWABUCHI Yu(u)ki, Aug 01 2012 *)

PROG

(PARI) A=vector(100); for(n=2, #A, A[n]=if(n%2, 3*A[n\2+1], A[n-1]+1)); A \\ Charles R Greathouse IV, Jul 24 2012

(PARI) is(n)=while(n, if(n%3>1, return(0)); n\=3); 1 \\ Charles R Greathouse IV, Mar 07 2013

(Haskell)

a005836 n = a005836_list !! (n-1)

a005836_list = filter ((== 1) . a039966) [0..]

-- Reinhard Zumkeller, Jun 09 2012, Sep 29 2011

CROSSREFS

a(n) = A005823(n)/2; a(n) = A003278(n)-1 = A033159(n)-2 = A033162(n)-3.

Cf. A005823, A032924, A054591, A007089, A081603, A081611, A000695, A007088, A033042-A033052, A074940, A083096. A002426, A003278, A004793, A055246, A062548, A081601, A089118, A121153, A170943, A185256.

For generating functions Prod_{k>=0} (1+a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.

Row 3 of array A104257.

Cf. A240075 (no 4-term starting at 0), A240556 (no 5-term starting at 0).

Sequence in context: A010388 A010400 A010439 * A054591 A121153 A242661

Adjacent sequences:  A005833 A005834 A005835 * A005837 A005838 A005839

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane, Jeffrey Shallit

EXTENSIONS

More terms from Emeric Deutsch and Bruce E. Sagan, Dec 04 2003

Offset corrected by N. J. A. Sloane, Mar 02 2008

Edited by the Associate Editors of the OEIS, Apr 07 2009

STATUS

approved

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Last modified October 30 07:54 EDT 2014. Contains 248796 sequences.