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A029744 Numbers of the form 2^n or 3*2^n. 65
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

WARNING: Several comments, formulas and programs seem to refer to the original version with offset=0. - M. F. Hasler, Oct 06 2014

Number of necklaces with n-1 beads and two colors that are the same when turned over and hence have reflection symmetry. [edited by Herbert Kociemba, Nov 24 2016]

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 greatest common divisor. - 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

Without initial 1, third row in array A228405. - Richard R. Forberg, Sep 06 2013

Numbers x such that sum_{i=1..k} (1/(p_i-1)) + product_{i=1..k} (1/(p_i-1)) is an integer, where p_i are the k prime factors of x (with multiplicity). In particular this sum is equal to n+1, being n the exponent of 2. [Paolo P. Lava, Feb 24 2014]

Also positions of records in A048673. A246360 gives the record values. - Antti Karttunen, Sep 23 2014

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

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

Index entries for linear recurrences with constant coefficients, signature (0,2).

Index entries for sequences related to necklaces

FORMULA

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 [Seems to refer to the original version with offset=0. - M. F. Hasler, Oct 06 2014]

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 Lafont, May 21 2008

For n=>3, a(n) = sqrt(2*a(n-1)^2 + (-2)^(n-3)). - Richard R. Forberg, Aug 20 2013

a(n) = A064216(A246360(n)). - Antti Karttunen, Sep 23 2014

a(n) = sqrt((17-(-1)^n)*2^(n-4)) for n>=2. - Anton Zakharov, Jul 24 2016

MAPLE

with(numtheory); P:=proc(q) local a, b, c, i, n;

for n from 1 to q do a:=ifactors(n)[2]; b:=add(a[i, 2]/(a[i, 1]-1), i=1..nops(a));

c:=mul((1/(a[i, 1]-1))^a[i, 2], i=1..nops(a)); if type(b+c, integer) then print(n);

fi; od; end: P(10^6); # Paolo P. Lava, Feb 24 2014

# Alternative:

1, seq(op([2^i, 3*2^(i-1)]), i=1..100); # Robert Israel, Sep 23 2014

MATHEMATICA

CoefficientList[Series[(-x^2 - 2*x - 1)/(2*x^2 - 1), {x, 0, 200}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)

Function[w, DeleteCases[Union@ Flatten@ w, k_ /; k > Max@ First@ w]]@ TensorProduct[{1, 3}, 2^Range[0, 22]] (* Michael De Vlieger, Nov 24 2016 *)

PROG

(PARI) a(n)=2^(n\2)*if(n%2, 2, 3/2) \\ Refers to the original version with offset=0. - M. F. Hasler, Oct 06 2014

(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

(Scheme) (define (A029744 n) (cond ((<= n 1) n) ((even? n) (expt 2 (/ n 2))) (else (* 3 (expt 2 (/ (- n 3) 2)))))) ;; Antti Karttunen, Sep 23 2014

CROSSREFS

Cf. A056493, A038754, A063759. Union of A000079 and A007283.

First differences are in A016116(n-1).

Cf. A082125, A094958, A048739, A048985, A193652, A048673, A064216, A246360.

Row sums of the triangle in sequence A119963. - John P. McSorley, Aug 31 2010

Sequence in context: A064428 A052810 A164090 * A018635 A018425 A018328

Adjacent sequences:  A029741 A029742 A029743 * A029745 A029746 A029747

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Corrected and extended by Joe Keane (jgk(AT)jgk.org), Feb 20 2000

STATUS

approved

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Last modified March 24 23:06 EDT 2017. Contains 284035 sequences.