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%I M2561 #240 Mar 09 2024 11:45:06
%S 3,6,12,24,48,96,192,384,768,1536,3072,6144,12288,24576,49152,98304,
%T 196608,393216,786432,1572864,3145728,6291456,12582912,25165824,
%U 50331648,100663296,201326592,402653184,805306368,1610612736,3221225472,6442450944,12884901888
%N a(n) = 3*2^n.
%C Same as Pisot sequences E(3, 6), L(3, 6), P(3, 6), T(3, 6). See A008776 for definitions of Pisot sequences.
%C Numbers k such that A006530(A000010(k)) = A000010(A006530(k)) = 2. - _Labos Elemer_, May 07 2002
%C Also least number m such that 2^n is the smallest proper divisor of m which is also a suffix of m in binary representation, see A080940. - _Reinhard Zumkeller_, Feb 25 2003
%C Length of the period of the sequence Fibonacci(k) (mod 2^(n+1)). - _Benoit Cloitre_, Mar 12 2003
%C The sequence of first differences is this sequence itself. - _Alexandre Wajnberg_ and _Eric Angelini_, Sep 07 2005
%C Subsequence of A122132. - _Reinhard Zumkeller_, Aug 21 2006
%C Apart from the first term, a subsequence of A124509. - _Reinhard Zumkeller_, Nov 04 2006
%C Total number of Latin n-dimensional hypercubes (Latin polyhedra) of order 3. - Kenji Ohkuma (k-ookuma(AT)ipa.go.jp), Jan 10 2007
%C Number of different ternary hypercubes of dimension n. - Edwin Soedarmadji (edwin(AT)systems.caltech.edu), Dec 10 2005
%C For n >= 1, a(n) is equal to the number of functions f:{1, 2, ..., n + 1} -> {1, 2, 3} such that for fixed, different x_1, x_2,...,x_n in {1, 2, ..., n + 1} and fixed y_1, y_2,...,y_n in {1, 2, 3} we have f(x_i) <> y_i, (i = 1,2,...,n). - _Milan Janjic_, May 10 2007
%C a(n) written in base 2: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, n times 0 (see A003953). - _Jaroslav Krizek_, Aug 17 2009
%C Subsequence of A051916. - _Reinhard Zumkeller_, Mar 20 2010
%C Numbers containing the number 3 in their Collatz trajectories. - _Reinhard Zumkeller_, Feb 20 2012
%C a(n-1) gives the number of ternary numbers with n digits with no two adjacent digits in common; e.g., for n=3 we have 010, 012, 020, 021, 101, 102, 120, 121, 201, 202, 210 and 212. - _Jon Perry_, Oct 10 2012
%C If n > 1, then a(n) is a solution for the equation sigma(x) + phi(x) = 3x-4. This equation also has solutions 84, 3348, 1450092, ... which are not of the form 3*2^n. - _Farideh Firoozbakht_, Nov 30 2013
%C a(n) is the upper bound for the "X-ray number" of any convex body in E^(n + 2), conjectured by Bezdek and Zamfirescu, and proved in the plane E^2 (see the paper by Bezdek and Zamfirescu). - _L. Edson Jeffery_, Jan 11 2014
%C If T is a topology on a set V of size n and T is not the discrete topology, then T has at most 3 * 2^(n-2) many open sets. See Brown and Stephen references. - _Ross La Haye_, Jan 19 2014
%C Comment from Charles Fefferman, courtesy of _Doron Zeilberger_, Dec 02 2014: (Start)
%C Fix a dimension n. For a real-valued function f defined on a finite set E in R^n, let Norm(f, E) denote the inf of the C^2 norms of all functions F on R^n that agree with f on E. Then there exist constants k and C depending only on the dimension n such that Norm(f, E) <= C*max{ Norm(f, S) }, where the max is taken over all k-point subsets S in E. Moreover, the best possible k is 3 * 2^(n-1).
%C The analogous result, with the same k, holds when the C^2 norm is replaced, e.g., by the C^1, alpha norm (0 < alpha <= 1). However, the optimal analogous k, e.g., for the C^3 norm is unknown.
%C For the above results, see Y. Brudnyi and P. Shvartsman (1994). (End)
%C Also, coordination sequence for (infinity, infinity, infinity) tiling of hyperbolic plane. - _N. J. A. Sloane_, Dec 29 2015
%C The average of consecutive powers of 2 beginning with 2^1. - _Melvin Peralta_ and Miriam Ong Ante, May 14 2016
%C For n > 1, a(n) is the smallest Zumkeller number with n divisors that are also Zumkeller numbers (A083207). - _Ivan N. Ianakiev_, Dec 09 2016
%C Also, for n >= 2, the number of length-n strings over the alphabet {0,1,2,3} having only the single letters as nonempty palindromic subwords. (Corollary 21 in Fleischer and Shallit) - _Jeffrey Shallit_, Dec 02 2019
%C Also, a(n) is the minimum link-length of any covering trail, circuit, path, and cycle for the set of the 2^(n+2) vertices of an (n+2)-dimensional hypercube. - _Marco Ripà_, Aug 22 2022
%C The known fixed points of maps n -> A163511(n) and n -> A243071(n). [See comments in A163511]. - _Antti Karttunen_, Sep 06 2023
%C The finite subsequence a(3), a(4), a(5), a(6) = 24, 48, 96, 192 is one of only two geometric sequences that can be formed with all interior angles (all integer, in degrees) of a simple polygon. The other sequence is a subsequence of A000244 (see comment there). - _Felix Huber_, Feb 15 2024
%D Jason I. Brown, Discrete Structures and Their Interactions, CRC Press, 2013, p. 71.
%D T. Ito, Method, equipment, program and storage media for producing tables, Publication number JP2004-272104A, Japan Patent Office (written in Japanese, a(2)=12, a(3)=24, a(4)=48, a(5)=96, a(6)=192, a(7)=384 (a(7)=284 was corrected)).
%D Kenji Ohkuma, Atsuhiro Yamagishi and Toru Ito, Cryptography Research Group Technical report, IT Security Center, Information-Technology Promotion Agency, JAPAN.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A007283/b007283.txt">Table of n, a(n) for n = 0..1000</a>
%H K. Bezdek and Tudor Zamfirescu, <a href="http://tzamfirescu.tricube.de/TZamfirescu-125.pdf">A Characterization of 3-dimensional Convex Sets with an Infinite X-ray Number</a>, in: Coll. Math. Soc. J. Bolyai 63., Intuitive Geometry, Szeged (Hungary), North-Holland, Amsterdam, 1991, pp. 33-38.
%H Yuri Brudnyi and Pavel Shvartsman, <a href="https://doi.org/10.1155/S1073792894000140">Generalizations of Whitney's extension theorem</a>, International Mathematics Research Notices 1994.3 (1994): 129-139.
%H J. W. Cannon and P. Wagreich, <a href="http://dx.doi.org/10.1007/BF01444714">Growth functions of surface groups</a>, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
%H Tomislav Došlić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Doslic/doslic3.html">Kepler-Bouwkamp Radius of Combinatorial Sequences</a>, Journal of Integer Sequences, Vol. 17, 2014, #14.11.3.
%H John Elias, <a href="/A007283/a007283.png">Illustration: 2^n+1 hexagram perimeters</a>
%H Lukas Fleischer and Jeffrey Shallit, <a href="https://arxiv.org/abs/1911.12464">Words With Few Palindromes, Revisited</a>, arxiv preprint arXiv:1911.12464 [cs.FL], November 27 2019.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Roberto Rinaldi and Marco Ripà, <a href="https://arxiv.org/abs/2212.11216">Optimal cycles enclosing all the nodes of a k-dimensional hypercube</a>, arXiv:2212.11216 [math.CO], 2022.
%H Edwin Soedarmadji, <a href="http://dx.doi.org/10.1016/j.disc.2006.02.011">Latin Hypercubes and MDS Codes</a>, Discrete Mathematics, Volume 306, Issue 12, Jun 28 2006, Pages 1232-1239
%H D. Stephen, <a href="http://www.jstor.org/stable/2315186">Topology on Finite Sets</a>, American Mathematical Monthly, 75: 739 - 741, 1968.
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (2).
%F G.f.: 3/(1-2*x).
%F a(n) = 2*a(n - 1), n > 0; a(0) = 3.
%F a(n) = Sum_{k = 0..n} (-1)^(k reduced (mod 3))*binomial(n, k). - _Benoit Cloitre_, Aug 20 2002
%F a(n) = A118416(n + 1, 2) for n > 1. - _Reinhard Zumkeller_, Apr 27 2006
%F a(n) = A000079(n) + A000079(n + 1). - _Zerinvary Lajos_, May 12 2007
%F a(n) = A000079(n)*3. - _Omar E. Pol_, Dec 16 2008
%F From _Paul Curtz_, Feb 05 2009: (Start)
%F a(n) = b(n) + b(n+3) for b = A001045, A078008, A154879.
%F a(n) = abs(b(n) - b(n+3)) with b(n) = (-1)^n*A084247(n). (End)
%F a(n) = 2^n + 2^(n + 1). - _Jaroslav Krizek_, Aug 17 2009
%F a(n) = A173786(n + 1, n) = A173787(n + 2, n). - _Reinhard Zumkeller_, Feb 28 2010
%F A216022(a(n)) = 6 and A216059(a(n)) = 7, for n > 0. - _Reinhard Zumkeller_, Sep 01 2012
%F a(n) = (A000225(n) + 1)*3. - _Martin Ettl_, Nov 11 2012
%F E.g.f.: 3*exp(2*x). - _Ilya Gutkovskiy_, May 15 2016
%F A020651(a(n)) = 2. - _Yosu Yurramendi_, Jun 01 2016
%F a(n) = sqrt(A014551(n + 1)*A014551(n + 2) + A014551(n)^2). - _Ezhilarasu Velayutham_, Sep 01 2019
%F a(A048672(n)) = A225546(A133466(n)). - _Michel Marcus_ and _Peter Munn_, Nov 29 2019
%F Sum_{n>=1} 1/a(n) = 2/3. - _Amiram Eldar_, Oct 28 2020
%p A007283:=n->3*2^n; seq(A007283(n), n=0..50); # _Wesley Ivan Hurt_, Dec 03 2013
%t Table[3(2^n), {n, 0, 32}] (* _Alonso del Arte_, Mar 24 2011 *)
%o (PARI) a(n)=3*2^n
%o (PARI) a(n)=3<<n \\ _Charles R Greathouse IV_, Oct 10 2012
%o (Magma) [3*2^n: n in [0..30]]; // _Vincenzo Librandi_, May 18 2011
%o (Haskell)
%o a007283 = (* 3) . (2 ^)
%o a007283_list = iterate (* 2) 3
%o -- _Reinhard Zumkeller_, Mar 18 2012, Feb 20 2012
%o (Maxima) A007283(n):=3*2^n$
%o makelist(A007283(n),n,0,30); /* _Martin Ettl_, Nov 11 2012 */
%o (Scala) (List.fill(40)(2: BigInt)).scanLeft(1: BigInt)(_ * _).map(3 * _) // _Alonso del Arte_, Nov 28 2019
%o (Python)
%o def A007283(n): return 3<<n # _Chai Wah Wu_, Feb 14 2023
%Y Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.
%Y Subsequence of the following sequences: A029744, A029747, A029748, A029750, A362804 (after 3), A364494, A364496, A364289, A364291, A364292, A364295, A364497, A364964, A365422.
%Y Essentially same as A003945 and A042950.
%Y Row sums of (5, 1)-Pascal triangle A093562 and of (1, 5) Pascal triangle A096940.
%Y Cf. A000079, A100540, A124508, A221718.
%Y Cf. Latin squares: A000315, A002860, A003090, A040082, A003191; Latin cubes: A098843, A098846, A098679, A099321.
%Y Cf. A133466, A225546, A163511, A243071.
%K easy,nonn
%O 0,1
%A _N. J. A. Sloane_
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