A007283 a(n) = 3*2^n.

3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944, 12884901888

Offset: 0

Author

N. J. A. Sloane

Keywords

• easy
• nonn

Comments

Same as Pisot sequences E(3, 6), L(3, 6), P(3, 6), T(3, 6). See A008776 for definitions of Pisot sequences.

Numbers k such that A006530(A000010(k)) = A000010(A006530(k)) = 2. - Labos Elemer, May 07 2002

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

Length of the period of the sequence Fibonacci(k) (mod 2^(n+1)). - Benoit Cloitre, Mar 12 2003

The sequence of first differences is this sequence itself. - Alexandre Wajnberg and Eric Angelini, Sep 07 2005

Subsequence of A122132. - Reinhard Zumkeller, Aug 21 2006

Apart from the first term, a subsequence of A124509. - Reinhard Zumkeller, Nov 04 2006

Total number of Latin n-dimensional hypercubes (Latin polyhedra) of order 3. - Kenji Ohkuma (k-ookuma(AT)ipa.go.jp), Jan 10 2007

Number of different ternary hypercubes of dimension n. - Edwin Soedarmadji (edwin(AT)systems.caltech.edu), Dec 10 2005

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

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

Subsequence of A051916. - Reinhard Zumkeller, Mar 20 2010

Numbers containing the number 3 in their Collatz trajectories. - Reinhard Zumkeller, Feb 20 2012

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

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

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

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

Comment from Charles Fefferman, courtesy of Doron Zeilberger, Dec 02 2014: (Start)

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).

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.

For the above results, see Y. Brudnyi and P. Shvartsman (1994). (End)

Also, coordination sequence for (infinity, infinity, infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015

The average of consecutive powers of 2 beginning with 2^1. - Melvin Peralta and Miriam Ong Ante, May 14 2016

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

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

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

The known fixed points of maps n -> A163511(n) and n -> A243071(n). [See comments in A163511]. - Antti Karttunen, Sep 06 2023

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

A level 1 Sierpiński triangle is a triangle. Level n+1 is formed from three copies of level n by identifying pairs of corner vertices of each pair of triangles. For n>2, a(n-3) is the radius of the level n Sierpiński triangle graph. - Allan Bickle, Aug 03 2024

References

• Jason I. Brown, Discrete Structures and Their Interactions, CRC Press, 2013, p. 71.
• 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)).
• Kenji Ohkuma, Atsuhiro Yamagishi and Toru Ito, Cryptography Research Group Technical report, IT Security Center, Information-Technology Promotion Agency, JAPAN.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• K. Bezdek and Tudor Zamfirescu, A Characterization of 3-dimensional Convex Sets with an Infinite X-ray Number, in: Coll. Math. Soc. J. Bolyai 63., Intuitive Geometry, Szeged (Hungary), North-Holland, Amsterdam, 1991, pp. 33-38.
• Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
• Yuri Brudnyi and Pavel Shvartsman, Generalizations of Whitney's extension theorem, International Mathematics Research Notices 1994.3 (1994): 129-139.
• J. W. Cannon and P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
• Tomislav Došlić, Kepler-Bouwkamp Radius of Combinatorial Sequences, Journal of Integer Sequences, Vol. 17, 2014, #14.11.3.
• John Elias, Illustration: 2^n+1 hexagram perimeters
• Lukas Fleischer and Jeffrey Shallit, Words With Few Palindromes, Revisited, arxiv preprint arXiv:1911.12464 [cs.FL], November 27 2019.
• A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
• Tanya Khovanova, Recursive Sequences
• Roberto Rinaldi and Marco Ripà, Optimal cycles enclosing all the nodes of a k-dimensional hypercube, arXiv:2212.11216 [math.CO], 2022.
• Edwin Soedarmadji, Latin Hypercubes and MDS Codes, Discrete Mathematics, Volume 306, Issue 12, Jun 28 2006, Pages 1232-1239
• D. Stephen, Topology on Finite Sets, American Mathematical Monthly, 75: 739 - 741, 1968.
• Index entries for linear recurrences with constant coefficients, signature (2).

Crossrefs

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.

Subsequence of the following sequences: A029744, A029747, A029748, A029750, A362804 (after 3), A364494, A364496, A364289, A364291, A364292, A364295, A364497, A364964, A365422.

Essentially same as A003945 and A042950.

Row sums of (5, 1)-Pascal triangle A093562 and of (1, 5) Pascal triangle A096940.

Cf. A000079, A100540, A124508, A221718.

Cf. Latin squares: A000315, A002860, A003090, A040082, A003191; Latin cubes: A098843, A098846, A098679, A099321.

Cf. A133466, A225546, A163511, A243071.

Cf. A007283, A029858, A067771, A193277, A233774, A233775, A246959.

Programs

Haskell

a007283 = (* 3) . (2 ^)
a007283_list = iterate (* 2) 3
-- Reinhard Zumkeller, Mar 18 2012, Feb 20 2012

Magma

[3*2^n: n in [0..30]]; // Vincenzo Librandi, May 18 2011

Maple

A007283:=n->3*2^n; seq(A007283(n), n=0..50); # Wesley Ivan Hurt, Dec 03 2013

Mathematica

Table[3(2^n), {n, 0, 32}] (* Alonso del Arte, Mar 24 2011 *)

Maxima

A007283(n):=3*2^n$
makelist(A007283(n),n,0,30); /* Martin Ettl, Nov 11 2012 */

PARI

a(n)=3*2^n

PARI

a(n)=3<Charles R Greathouse IV, Oct 10 2012

Python

def A007283(n): return 3<Chai Wah Wu, Feb 14 2023

Scala

(List.fill(40)(2: BigInt)).scanLeft(1: BigInt)( * ).map(3 * ) // _Alonso del Arte, Nov 28 2019

Formula

G.f.: 3/(1-2*x).

a(n) = 2*a(n - 1), n > 0; a(0) = 3.

a(n) = Sum_{k = 0..n} (-1)^(k reduced (mod 3))*binomial(n, k). - Benoit Cloitre, Aug 20 2002

a(n) = A118416(n + 1, 2) for n > 1. - Reinhard Zumkeller, Apr 27 2006

a(n) = A000079(n) + A000079(n + 1). - Zerinvary Lajos, May 12 2007

a(n) = A000079(n)*3. - Omar E. Pol, Dec 16 2008

From Paul Curtz, Feb 05 2009: (Start)

a(n) = b(n) + b(n+3) for b = A001045, A078008, A154879.

a(n) = abs(b(n) - b(n+3)) with b(n) = (-1)^n*A084247(n). (End)

a(n) = 2^n + 2^(n + 1). - Jaroslav Krizek, Aug 17 2009

a(n) = A173786(n + 1, n) = A173787(n + 2, n). - Reinhard Zumkeller, Feb 28 2010

A216022(a(n)) = 6 and A216059(a(n)) = 7, for n > 0. - Reinhard Zumkeller, Sep 01 2012

a(n) = (A000225(n) + 1)*3. - Martin Ettl, Nov 11 2012

E.g.f.: 3*exp(2*x). - Ilya Gutkovskiy, May 15 2016

A020651(a(n)) = 2. - Yosu Yurramendi, Jun 01 2016

a(n) = sqrt(A014551(n + 1)*A014551(n + 2) + A014551(n)^2). - Ezhilarasu Velayutham, Sep 01 2019

a(A048672(n)) = A225546(A133466(n)). - Michel Marcus and Peter Munn, Nov 29 2019

Sum_{n>=1} 1/a(n) = 2/3. - Amiram Eldar, Oct 28 2020