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A027649 2*(3^n)-2^n. 24
1, 4, 14, 46, 146, 454, 1394, 4246, 12866, 38854, 117074, 352246, 1058786, 3180454, 9549554, 28665046, 86027906, 258149254, 774578834, 2323998646, 6972520226, 20918609254, 62757924914, 188277969046 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Poly-Bernoulli numbers B_n^(k) with k=-2.

Binomial transform of A007051, if both sequences start at 0. Binomial transform of A000225(n+1). - Paul Barry, Mar 24 2003

Euler expands (1-z)/(1-5z+6z^2) and finds the general term. Section 226 of the Introductio indicates that he could have written down the recursion relation: a(n) = 5 a(n-1)-6 a(n-2). - V. Frederick Rickey (fred-rickey(AT)usma.edu), Feb 10 2006

Let R 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), xRy if x is a subset of y or y is a subset of x. Then a(n) = |R|. - Ross La Haye, Dec 22 2006 Ross

If x, y are two n-bit binary strings then a(n) gives the number of pairs (x,y) such that XOR(x, y) = ABS(x - y) [From Ramasamy Chandramouli, Feb 15 2009]

Equals row sums of the triangular version of A038573. [From Gary W. Adamson, Jun 04 2009]

Inverse binomial transform of A085350. [From Paul Curtz, Nov 14 2009]

Related to the number of even a's in a non-trivial cycle (should one exist) in the 3x+1 Problem, where a <= floor( log(2*(3^n) - 2^n) / log 2). The value n correlates to the number of odds in such a non-trivial cycle. See page 1288 of Crandall's paper. Also, this relation gives another proof that the number of odds divided by the number of evens in a non-trivial cycle is bounded by log 2 / log 3 (this observation does not resolve the finite cycles conjecture as the value could be arbitrarily close to this bound). However, the same argument gives that log 2 / log 3 is less than or equal to the number of odds divided by the number of evens in a divergent sequence (should one exist), as log 2 / log 3 is the limit value for a cycle of an arbitrarily large length, where the length is given by the value n. [From Jeffrey R. Goodwin, Aug 04 2011]

REFERENCES

R. E. Crandall, On the "3x+1" Problem, Math. Comp. 32 (1978), 1281-1292.

Leonhard Euler, Introduction in analysin infinitorum (1748), section 216.

Ken Kamano, Sums of Products of Poly-Bernoulli Numbers of Negative Index, Journal of Integer Sequences, Vol. 15 (2012), #12.1.3.

M. Kaneko, Poly-Bernoulli numbers, J. Theorie des Nombres Bordeaux 9 (1997), 221-228.

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.

LINKS

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

M. Kaneko, Poly-Bernoulli numbers

Index entries for sequences related to Bernoulli numbers.

Index to sequences with linear recurrences with constant coefficients, signature (5,-6).

FORMULA

G.f.: (1-x)/((1-2*x)*(1-3*x)). a(0)=1, a(n)=3*a(n-1)+2^(n-1).

a(n)=sum{k=0..n, C(n, k)(2^(k + 1) - 1)}. - Paul Barry, Mar 24 2003

Partial sums of A053581. - Paul Barry, Jun 26 2003

Main diagonal of array (A085870) defined by T(i, 1)=2^i-1, T(1, j)=2^j-1, T(i, j)=T(i-1, j)+T(i-1, j-1). - Benoit Cloitre, Aug 05 2003

a(n) = A090888(n, 3). - Ross La Haye, Sep 21 2004

a(n)=sum{k=0..n, C(n+2, k+1)*sum{j=0..floor(k/2), A001045(k-2j)}} - Paul Barry, Apr 17 2005

a(n)=sum{k=0..n, sum{j=0..n, C(n,j)C(j+1,k+1)}}; - Paul Barry, Sep 18 2006

Row sums of triangle A131109. - Gary W. Adamson, Jun 15 2007

a(n)= A166060(n+1)/6. [Philippe Deléham, Oct 21 2009]

a(n) = 5a(n-1)-6a(n-2), a(0)=1, a(1)=4. [Harvey P. Dale, Apr 22 2011]

a(n) = A217764(n,2). - Ross La Haye, Mar 27 2013

MAPLE

(-1)^n*sum( (-1)^'m'*'m'!*stirling2(n, 'm')/('m'+1)^k, 'm'=0..n);

MATHEMATICA

Table[2(3^n)-2^n, {n, 0, 25}] (* or *)

LinearRecurrence[{5, -6}, {1, 4}, 25]  (* Harvey P. Dale, Apr 22 2011 *)

PROG

(PARI) a(n)=2*(3^n)-2^n \\ Charles R Greathouse IV, Jul 16, 2011

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

(Haskell)

a027649 n = a027649_list !! n

a027649_list = map fst $ iterate (\(u, v) -> (3 * u + v, 2 * v)) (1, 1)

-- Reinhard Zumkeller, Jun 09 2013

CROSSREFS

Row 2 of array A099594.

Cf. A131109.

Cf. A038573.

Cf. A001047.

Sequence in context: A029868 A030267 A026290 * A049221 A081670 A124805

Adjacent sequences:  A027646 A027647 A027648 * A027650 A027651 A027652

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Better formulae from David W. Wilson and Michael Somos.

Incorrect formula removed by Charles R Greathouse IV, Mar 18 2010

STATUS

approved

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Last modified October 30 11:52 EDT 2014. Contains 248800 sequences.