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 A029744 Numbers of the form 2^n or 3*2^n. 106
 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 This entry is a list, and so has offset 1. WARNING: However, in this entry several comments, formulas and programs seem to refer to the original version of this sequence which had 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 no fewer 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. - 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 Also positions of records in A048673. A246360 gives the record values. - Antti Karttunen, Sep 23 2014 Known in numerical mathematics as "Bulirsch sequence", used in various extrapolation methods for step size control. - Peter Luschny, Oct 30 2019 For n > 1, squares of the terms can be expressed as the sum of two powers of two: 2^x + 2^y. - Karl-Heinz Hofmann, Sep 08 2022 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..2000 Michael De Vlieger, Thomas Scheuerle, Rémy Sigrist, N. J. A. Sloane, and Walter Trump, The Binary Two-Up Sequence, arXiv:2209.04108 [math.CO], Sep 11 2022. David Eppstein, Making Change in 2048, arXiv:1804.07396 [cs.DM], 2018. Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy] John P. McSorley and Alan H. Schoen, Rhombic tilings of (n,k)-ovals, (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) = 2*a(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 Sum_{n>=1} 1/a(n) = 8/3. - Amiram Eldar, Nov 12 2020 a(n) = 2^(n/2) if n is even. a(n) = 3 * 2^((n-3)/2) if n is odd and for n>1. - Karl-Heinz Hofmann, Sep 08 2022 MAPLE 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 *) LinearRecurrence[{0, 2}, {1, 2, 3}, 50] (* Harvey P. Dale, Jul 04 2017 *) PROG (PARI) a(n)=if(n%2, 3/2, 2)<<((n-1)\2)\1 (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 (Python) def A029744(n): if n == 1: return 1 elif n % 2 == 0: return 2**(n//2) else: return 3 * 2**((n-3)//2) # Karl-Heinz Hofmann, Sep 08 2022 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, A354785. Row sums of the triangle in sequence A119963. - John P. McSorley, Aug 31 2010 The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent. There may be minor differences from (s(n)) at the start, and a shift of indices. A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A060482 (s(n)-3); A136252 (s(n)-3); A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A354785 (3*s(n)), A061776 (3*s(n)-6); A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022 Sequence in context: A320315 A364956 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 June 14 20:28 EDT 2024. Contains 373401 sequences. (Running on oeis4.)