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A007283 a(n) = 3*2^n.
(Formerly M2561)
157
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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 n such that P[phi[n]]=phi[P[n]], where P[x] is the largest prime-factor of x; A006530[A000010(n)]=A000010[A006530(n)]=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 first differences are the 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

3 times powers of 2. - Omar E. Pol, Dec 16 2008

From Jacobsthal numbers and successive differences. A001045=0,1,1,3,5,11,21,43,; A078008=1,0,2,2,6,10,22,42,; A084247 signed=-1,2,0,4,4,12,20,44,; A154879=3,-2,4,0,8,8,24,40,. For every sequence k(n), a(n)= k(n)+k(n+3).Then A001045(n)+A001045(n+3), A078008(n)+A078008(n+3), .. . - Paul Curtz, Feb 05 2009

a(n) written in base 2: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, n times 0 (see A003953(n)). - Jaroslav Krizek, Aug 17 2009

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

Subsequence of A051916. - Reinhard Zumkeller, Mar 20 2010

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

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

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

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 T. 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.

Yuri Brudnyi, and Pavel Shvartsman, Generalizations of Whitney's extension theorem, International Mathematics Research Notices 1994.3 (1994): 129-139.

J. W. Cannon, P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.

T. Doslic, Kepler-Bouwkamp Radius of Combinatorial Sequences, Journal of Integer Sequences, Vol. 17, 2014, #14.11.3.

Tanya Khovanova, Recursive Sequences

E. 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).

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))*C(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

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

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

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

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 *)

PROG

(PARI) a(n)=3*2^n

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

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

(Haskell)

a007283 = (* 3) . (2 ^)

a007283_list = iterate (* 2) 3

-- Reinhard Zumkeller, Mar 18 2012, Feb 20 2012

(Maxima) A007283(n):=3*2^n$

makelist(A007283(n), n, 0, 30); /* Martin Ettl, Nov 11 2012 */

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 A029744.

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.

Sequence in context: A170636 A170684 A003945 * A049942 A200463 A099844

Adjacent sequences:  A007280 A007281 A007282 * A007284 A007285 A007286

KEYWORD

easy,nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 23 13:38 EDT 2017. Contains 285322 sequences.