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A029744
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Numbers of the form 2^n or 3*2^n.
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31
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1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072, 196608, 262144, 393216, 524288, 786432, 1048576, 1572864, 2097152, 3145728, 4194304
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OFFSET
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1,2
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COMMENTS
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Necklaces with n beads that are the same when turned over.
The subset {a(1),...,a(2k)} contains all proper divisors of 3*2^k. - Ralf Stephan, Jun 02 2003
Let k = any nonnegative integer and j = 0 or 1. Then n+1 = 2k + 3j and a(n) = 2^k*3^j. - Andras Erszegi (erszegi.andras(AT)chello.hu), Jul 30 2005
Smallest number having not less prime factors than any predecessor, a(0)=1; A110654(n)=A001222(a(n)); complement of A116451. - Reinhard Zumkeller, Feb 16 2006
A093873(a(n)) = 1. - Reinhard Zumkeller, Oct 13 2006
a(n)=a(n-1)+a(n-2)-GCD(a(n-1),a(n-2)),n>=3, a(1)=2,a(2)=3,GCD gratest common divisor. [From Ctibor O. Zizka, Jun 06 2009]
Where records occur in A048985: A193652(n) = A048985(a(n)) and A193652(n) < A048985(m) for m < a(n). [Reinhard Zumkeller, Aug 08 2011]
A002348(a(n)) = A000079(n-3) for n > 2. [Reinhard Zumkeller, Mar 18 2012]
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REFERENCES
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John P. McSorley and Alan H. Schoen: On k-Ovals and (n, k, lambda)-Cyclic Difference Sets, and Related Topics, Discrete Math., 313 (2013), 129-154. - From N. J. A. Sloane, Nov 26 2012
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Guo-Niu Han, Enumeration of Standard Puzzles
Index to sequences with linear recurrences with constant coefficients, signature (0,2).
Index entries for sequences related to necklaces
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FORMULA
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a(n)=2*A000029(n)-A000031(n).
For n>2 a(n)=2a(n-2); for n>3 a(n)=a(n-1)*a(n-2)/a(n-3). G.f.: (1+x)^2/(1-2*x^2). - Henry Bottomley, Jul 15 2001, corrected May 04 2007
a(0)=1, a(1)=1 and a(n) = a(n-2) * ( floor(a(n-1)/a(n-2)) + 1 ) - Benoit Cloitre, Aug 13 2002
(3/4+sqrt(1/2))*sqrt(2)^n + (3/4-sqrt(1/2))*(-sqrt(2))^n. a(0)=1, a(2n) = a(n-1)*a(n), a(2n+1) = a(n) + 2^floor((n-1)/2). Ralf Stephan, Apr 16 2003
Binomial transform is A048739. - Paul Barry, Apr 23 2004
E.g.f.: (cosh(x/sqrt(2))+sqrt(2)sinh(x/sqrt(2)))^2.
a(1) = 1; a(n+1) = a(n) + A000010(a(n)) - Stefan Steinerberger, Dec 20 2007
u(2)=1, v(2)=1, u(n)=2*v(n-1), v(n)=u(n-1), a(n)=u(n)+v(n) - Jaume Oliver (joliverlafont(AT)gmail.com), May 21 2008
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MATHEMATICA
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CoefficientList[Series[(-x^2 - 2*x - 1)/(2*x^2 - 1), {x, 0, 200}], x] (* From Vladimir Joseph Stephan Orlovsky, June 10 2011 *)
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PROG
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(PARI) a(n)=if(n<1, n==0, 2^(n\2)*if(n%2, 2, 3/2))
(Haskell)
a029744 n = a029744_list !! (n-1)
a029744_list = 1 : iterate
(\x -> if x `mod` 3 == 0 then 4 * x `div` 3 else 3 * x `div` 2) 2
-- Reinhard Zumkeller, Mar 18 2012
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CROSSREFS
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Cf. A056493, A038754, A063759. Union of A000079 and A007283.
First differences are in A016116(n-1).
Cf. A082125, A094958.
Row sums of the triangle in sequence A119963. [From John P. McSorley, Aug 31 2010]
Sequence in context: A052810 A079647 A164090 * A018635 A018425 A018328
Adjacent sequences: A029741 A029742 A029743 * A029745 A029746 A029747
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Corrected and extended by Joe Keane (jgk(AT)jgk.org), Feb 20 2000
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STATUS
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approved
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