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

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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
• Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024. See pp. 11, 22.
• 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

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

Programs

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

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 *)

PARI

a(n)=if(n%2,3/2,2)<<((n-1)\2)\1

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

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

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

Extensions

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