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A001047 a(n) = 3^n - 2^n.
(Formerly M3887 N1596)
128
0, 1, 5, 19, 65, 211, 665, 2059, 6305, 19171, 58025, 175099, 527345, 1586131, 4766585, 14316139, 42981185, 129009091, 387158345, 1161737179, 3485735825, 10458256051, 31376865305, 94134790219, 282412759265, 847255055011, 2541798719465, 7625463267259, 22876524019505 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) = sum of the elements in the n-th row of triangle pertaining to A036561. - Amarnath Murthy, Jan 02 2002

Number of 2 X n binary arrays with a path of adjacent 1's and no path of adjacent 0's from top row to bottom row. - R. H. Hardin, Mar 21 2002

With offset 1, partial sums of A027649. - Paul Barry, Jun 24 2003

Number of distinct lines through the origin in the n-dimensional lattice of side length 2. A049691 has the values for the 2-dimensional lattice of side length n. - Joshua Zucker, Nov 19 2003

a(n) = A083323(n)-1 = A056182(n)/2 = (A002783(n)-1)/2 = (A003063(n+2)-A003063(n+1))/2. - Ralf Stephan, Jan 12 2004

a(n+1)/(n+1)=(3*3^n-2*2^n)/(n+1) is the second binomial transform of the harmonic sequence 1/(n+1). - Paul Barry, Apr 19 2005

a(n) = A112626(n, 1). - Ross La Haye, Jan 11 2006

a(n+1) = sums of n-th row of A036561. - Reinhard Zumkeller, May 14 2006

The sequence gives the sum of the lengths of the segments in Cantor's dust generating sequence up to the i-th step. Measurement unit = length of the segment of i-th step. - Giorgio Balzarotti, Nov 18 2006

Let T be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xTy if x is a proper subset of y. Then a(n) = |T|. - Ross La Haye, Dec 22 2006

From Alexander Adamchuk, Jan 04 2007: (Start)

a(n) is prime for n = {2,3,5,17,29,31,53,59,101,277,647,1061,2381,...} = A057468(n) Numbers n such that 3^n - 2^n is prime.

p divides a(p) - 1 for prime p.

Quotients (3^p - 2^p - 1)/p, where p = prime(n), are listed in A127071 = {2,6,42,294,15918,122010,7588770,61144062,...}.

Numbers n such that n divides 3^n - 2^n - 1 are listed in A127072 = {1,2,3,4,5,7,8,9,11,13,16,17,19,23,27,29,31,32,37,41,43,45,47,49,53,59,61,64,67,71,73,79,81,83,89,97,...}.

Pseudoprimes in A127072(n) include all powers of primes {2,3,7} and some composite numbers that are listed in A127073(n) = {45,245,405,561,637,639,833,891,...}, which includes all Carmichael numbers A002997(n) = {561,1105,1729,2465,2821,6601,8911,10585,15841,29341,...}.

Numbers n such that n^2 divides 3^n - 2^n - 1 are listed in A127074(n) = {1,2,3,4,7,49,179,619,17807,...}.

5 divides a(2n).

5^2 divides a(2*5n).

5^3 divides a(2*5^2n).

5^4 divides a(2*5^3n).

7^2 divides a(6*7n).

13 divides a(4n).

13^2 divides a(4*13n).

19 divides a(3n).

19^2 divides a(3*19n).

23^2 divides a(11n).

23^3 divides a(11*23n).

23^4 divides a(11*23^2n).

29 divides a(7n).

p divides a((p-1)n) for prime p>3.

p divides a((p-1)/2)) for prime p = {5,19,23,29,43,47,53,...} = A097936(n) Primes p such that p divides 3^((p-1)/2) - 2^((p-1)/2). Also primes p such that 6 is a square mod p, except {2,3}, A038876(n).

p^(k+1) divides a(p^k*(p-1)/2*n) for prime p = {5,19,23,29,43,47,53,...} = A097936(n).

p^(k+1) divides a(p^k*(p-1)*n) for prime p>3.

Note the exception that for p = 23, p^(k+2) divides a(p^k*(p-1)/2*n).

There are no more such exceptions for primes p up to 600000. (End)

Final digits of terms follow sequence 1,5,9,5. - Enoch Haga, Nov 26 2007

This is also the second column sequence of the Sheffer triangle A143494 (2-restricted Stirling2 numbers). See the e.g.f. given below. - Wolfdieter Lang, Oct 08 2011

a(n) = sum_{k=0..2} Stirling1(2,k)*(k+1)^n = c_2^{(-n)}, poly-Cauchy numbers. - Takao Komatsu, Mar 28 2013

a(n) = A227048(n,A098294(n)). - Reinhard Zumkeller, Jun 30 2013

Partial sums give A000392. - Jon Perry, Apr 05 2014

For n >= 1, this is also row 2 of A281890: when consecutive positive integers are written as a product of primes in nondecreasing order, "3" occurs in n-th position a(n) times out of every 6^n. - Peter Munn, May 17 2017

REFERENCES

Archimedeans Problems Drive, Eureka, 24 (1961), 20.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..200

Nathan Bliss et al., Strong divisibility, cyclotomic polynomials and iterated polynomials, Am. Math. Monthly, 120 (2013), 519-536.

S. Giraudo, Combinatorial operads from monoids, arXiv preprint arXiv:1306.6938 [math.CO], 2013.

Samuele Giraudo, Pluriassociative algebras I: The pluriassociative operad, arXiv:1603.01040 [math.CO], 2016.

R. K. Guy, Letters to N. J. A. Sloane, June-August 1968

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 397

G. Kreweras, Inversion des polynomes de Bell bidimensionnels et application au dénombrement des relations binaires connexes, C. R. Acad. Sci. Paris Ser. A-B 268 1969 A577-A579.

Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.

R. Miles, Synchronization points and associated dynamical invariants

J. Perry, Relation to Collatz problem

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for linear recurrences with constant coefficients, signature (5,-6).

FORMULA

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

a(n) = 5*a(n-1) - 6*a(n-2).

a(n) = 3*a(n-1) + 2^(n-1). - Jon Perry, Aug 23, 2002

Starting 0, 0, 1, 5, 19, ... this is 3^n/3 - 2^n/2 + 0^n/6, the binomial transform of A086218. - Paul Barry, Aug 18 2003

Binomial transform of A000225. - Ross La Haye, Feb 07 2005

a(n) = Sum_{k=0..n-1} binomial(n, k)*2^k. - Ross La Haye, Aug 20 2005

a(n) = 2^(2n) - A083324(n). - Ross La Haye, Sep 10 2005

E.g.f.: exp(3*x) - exp(2*x). - Mohammad K. Azarian, Jan 14 2009

a(n) = A217764(n,1). - Ross La Haye, Mar 27 2013

a(n) = 2*a(n-1) + 3^(n-1). - Toby Gottfried, Mar 28 2013

a(n) = A000244(n) - A000079(n). - Omar E. Pol, Mar 28 2013

MAPLE

a(n)=seq(sum(2^i*3^(n-i), i=0..n), n=0..40); # Giorgio Balzarotti, Nov 18 2006

A001047:=1/(3*z-1)/(2*z-1); # Simon Plouffe in his 1992 dissertation, dropping the initial zero

MATHEMATICA

Table[ 3^n - 2^n, {n, 0, 25} ]

LinearRecurrence[{5, -6}, {0, 1}, 25] (* Harvey P. Dale, Aug 18 2011 *)

Numerator@NestList[(3#+1)/2&, 1/2, 100] (* Moshe Levin, Oct 03 2011 *)

PROG

(Python) [3**n - 2**n for n in range(25)] # Ross La Haye, Aug 19 2005; corrected by David Radcliffe, Jun 26 2016

(Sage) [lucas_number1(n, 5, 6) for n in xrange(0, 26)] # Zerinvary Lajos, Apr 22 2009

(PARI) {a(n) = 3^n - 2^n};

(MAGMA) [3^n - 2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011

(Haskell)

a001047 n = a001047_list !! n

a001047_list = map fst $ iterate (\(u, v) -> (3 * u + v, 2 * v)) (0, 1)

-- Reinhard Zumkeller, Jun 09 2013

CROSSREFS

Cf. A000225, A016189, A036561, A097936, A038876, A127071, A127072, A127073, A127074, A002997, A057468, A109235, A281890.

a(n) = row sums of A091913, row 2 of A047969, column 1 of A090888 and column 1 of A038719.

Cf. A000392, A240400.

Cf. partitions: A241766, A241759.

A diagonal of A262307.

Sequence in context: A229239 A001870 A025568 * A099448 A239618 A124806

Adjacent sequences:  A001044 A001045 A001046 * A001048 A001049 A001050

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, R. K. Guy

EXTENSIONS

Edited by Charles R Greathouse IV, Mar 24 2010

STATUS

approved

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Last modified June 22 12:24 EDT 2017. Contains 288613 sequences.