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A027649 a(n) = 2*(3^n) - 2^n. 30
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

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). - Ramasamy Chandramouli, Feb 15 2009

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

Inverse binomial transform of A085350. - 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(2*(3^n) - 2^n)). 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. - Jeffrey R. Goodwin, Aug 04 2011

Row sums of Riordan triangle A106516. - Wolfdieter Lang, Jan 09 2015

Number of restricted barred preferential arrangements having 3 bars in-which the sections are all restricted sections; such that (for fixed sections i and j) section i or section j is empty. - Sithembele Nkonkobe, Oct 12 2015

REFERENCES

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

LINKS

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

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

Shalosh B. Ekhad, N. J. A. Sloane, and  Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.

Shalosh B. Ekhad, N. J. A. Sloane, and  Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.

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

Masanobu Kaneko, Poly-Bernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 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.

S. Nkonkobe and V. Murali, On some properties and relations between restricted barred preferential arrangements, multi-poly-bernoulli numbers and related numbers, arXiv:1509.07352 [math.CO], 2015.

Index entries for sequences related to Bernoulli numbers.

Index entries for 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} binomial(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} binomial(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} binomial(n,j)*binomial(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

For n>0, a(n) = 3 * a(n-1) + 2^(n-1) = 2 * (a(n-1) + 3^(n-1)). - J. Conrad, Oct 29 2015

for n>0, a(n) = 2 * (1 + 2^(n-2) + Sum_{x=1..n-2}Sum_{k=0..x-1}(binomial(x-1,k)*(2^(k+1) + 2^(n-x+k)))). - J. Conrad, Dec 10 2015

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

(PARI) Vec((1-x)/((1-2*x)*(1-3*x)) + O(x^100)) \\ Altug Alkan, Oct 12 2015

CROSSREFS

Row n = 2 of array A099594.

Cf. A000079, A001047, A038573, A106516, A131109.

Sequence in context: A029868 A030267 A026290 * A049221 A081670 A258255

Adjacent sequences:  A027646 A027647 A027648 * A027650 A027651 A027652

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Better formulas 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 August 30 04:31 EDT 2016. Contains 275965 sequences.