cp's OEIS Frontend

A029744 Numbers of the form 2^n or 3*2^n.

%I A029744 #210 Jun 28 2024 01:26:23
%S A029744 1,2,3,4,6,8,12,16,24,32,48,64,96,128,192,256,384,512,768,1024,1536,
%T A029744 2048,3072,4096,6144,8192,12288,16384,24576,32768,49152,65536,98304,
%U A029744 131072,196608,262144,393216,524288,786432,1048576,1572864,2097152,3145728,4194304
%N A029744 Numbers of the form 2^n or 3*2^n.
%C A029744 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
%C A029744 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]
%C A029744 The subset {a(1),...,a(2k)} contains all proper divisors of 3*2^k. - _Ralf Stephan_, Jun 02 2003
%C A029744 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
%C A029744 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
%C A029744 A093873(a(n)) = 1. - _Reinhard Zumkeller_, Oct 13 2006
%C A029744 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
%C A029744 Where records occur in A048985: A193652(n) = A048985(a(n)) and A193652(n) < A048985(m) for m < a(n). - _Reinhard Zumkeller_, Aug 08 2011
%C A029744 A002348(a(n)) = A000079(n-3) for n > 2. - _Reinhard Zumkeller_, Mar 18 2012
%C A029744 Without initial 1, third row in array A228405. - _Richard R. Forberg_, Sep 06 2013
%C A029744 Also positions of records in A048673. A246360 gives the record values. - _Antti Karttunen_, Sep 23 2014
%C A029744 Known in numerical mathematics as "Bulirsch sequence", used in various extrapolation methods for step size control. - _Peter Luschny_, Oct 30 2019
%C A029744 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
%H A029744 Vincenzo Librandi, <a href="/A029744/b029744.txt">Table of n, a(n) for n = 1..2000</a>
%H A029744 Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, <a href="https://arxiv.org/abs/2406.12941">Metered Parking Functions</a>, arXiv:2406.12941 [math.CO], 2024. See pp. 11, 22.
%H A029744 Michael De Vlieger, Thomas Scheuerle, Rémy Sigrist, N. J. A. Sloane, and Walter Trump, <a href="http://arxiv.org/abs/2209.04108">The Binary Two-Up Sequence</a>, arXiv:2209.04108 [math.CO], Sep 11 2022.
%H A029744 David Eppstein, <a href="https://arxiv.org/abs/1804.07396">Making Change in 2048</a>, arXiv:1804.07396 [cs.DM], 2018.
%H A029744 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>. [Cached copy]
%H A029744 John P. McSorley and Alan H. Schoen, <a href="http://dx.doi.org/10.1016/j.disc.2012.08.021">Rhombic tilings of (n,k)-ovals, (n, k, lambda)-cyclic difference sets, and related topics</a>, Discrete Math., 313 (2013), 129-154. - From _N. J. A. Sloane_, Nov 26 2012
%H A029744 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,2).
%H A029744 <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>
%F A029744 a(n) = 2*A000029(n) - A000031(n).
%F A029744 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
%F A029744 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
%F A029744 (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]
%F A029744 Binomial transform is A048739. - _Paul Barry_, Apr 23 2004
%F A029744 E.g.f.: (cosh(x/sqrt(2)) + sqrt(2)sinh(x/sqrt(2)))^2.
%F A029744 a(1) = 1; a(n+1) = a(n) + A000010(a(n)). - _Stefan Steinerberger_, Dec 20 2007
%F A029744 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
%F A029744 For n => 3, a(n) = sqrt(2*a(n-1)^2 + (-2)^(n-3)). - _Richard R. Forberg_, Aug 20 2013
%F A029744 a(n) = A064216(A246360(n)). - _Antti Karttunen_, Sep 23 2014
%F A029744 a(n) = sqrt((17 - (-1)^n)*2^(n-4)) for n >= 2. - _Anton Zakharov_, Jul 24 2016
%F A029744 Sum_{n>=1} 1/a(n) = 8/3. - _Amiram Eldar_, Nov 12 2020
%F A029744 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
%p A029744 1,seq(op([2^i,3*2^(i-1)]),i=1..100); # _Robert Israel_, Sep 23 2014
%t A029744 CoefficientList[Series[(-x^2 - 2*x - 1)/(2*x^2 - 1), {x, 0, 200}], x] (* _Vladimir Joseph Stephan Orlovsky_, Jun 10 2011 *)
%t A029744 Function[w, DeleteCases[Union@ Flatten@ w, k_ /; k > Max@ First@ w]]@ TensorProduct[{1, 3}, 2^Range[0, 22]] (* _Michael De Vlieger_, Nov 24 2016 *)
%t A029744 LinearRecurrence[{0,2},{1,2,3},50] (* _Harvey P. Dale_, Jul 04 2017 *)
%o A029744 (PARI) a(n)=if(n%2,3/2,2)<<((n-1)\2)\1
%o A029744 (Haskell)
%o A029744 a029744 n = a029744_list !! (n-1)
%o A029744 a029744_list = 1 : iterate
%o A029744    (\x -> if x `mod` 3 == 0 then 4 * x `div` 3 else 3 * x `div` 2) 2
%o A029744 -- _Reinhard Zumkeller_, Mar 18 2012
%o A029744 (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
%o A029744 (Python)
%o A029744 def A029744(n):
%o A029744     if n == 1: return 1
%o A029744     elif n % 2 == 0: return 2**(n//2)
%o A029744     else: return 3 * 2**((n-3)//2) # _Karl-Heinz Hofmann_, Sep 08 2022
%Y A029744 Cf. A056493, A038754, A063759. Union of A000079 and A007283.
%Y A029744 First differences are in A016116(n-1).
%Y A029744 Cf. A082125, A094958, A048739, A048985, A193652, A048673, A064216, A246360, A354785.
%Y A029744 Row sums of the triangle in sequence A119963. - _John P. McSorley_, Aug 31 2010
%Y A029744 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
%K A029744 nonn,easy
%O A029744 1,2
%A A029744 _N. J. A. Sloane_
%E A029744 Corrected and extended by Joe Keane (jgk(AT)jgk.org), Feb 20 2000