

A027649


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


29



1, 4, 14, 46, 146, 454, 1394, 4246, 12866, 38854, 117074, 352246, 1058786, 3180454, 9549554, 28665046, 86027906, 258149254, 774578834, 2323998646, 6972520226, 20918609254, 62757924914, 188277969046
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OFFSET

0,2


COMMENTS

PolyBernoulli 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 (1z)/(15z+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(n1)6 a(n2).  V. Frederick Rickey (fredrickey(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 nbit 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 nontrivial 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 nontrivial 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 nontrivial 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


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) 12811292.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A MetaAlgorithm for Creating Fast Algorithms for Counting ON Cells in OddRule Cellular Automata, arXiv:1503.01796, 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, OddRule Cellular Automata on the Square Grid, arXiv:1503.04249, 2015.
Ken Kamano, Sums of Products of PolyBernoulli Numbers of Negative Index, Journal of Integer Sequences, Vol. 15 (2012), #12.1.3.
Masanobu Kaneko, PolyBernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221228.
Ross La Haye, Binary Relations on the Power Set of an nElement Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for sequences related to Bernoulli numbers.
Index to sequences with linear recurrences with constant coefficients, signature (5,6).


FORMULA

G.f.: (1x)/((12*x)*(13*x)). a(0)=1, a(n)=3*a(n1)+2^(n1).
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^i1, T(1, j)=2^j1, T(i, j)=T(i1, j)+T(i1, j1).  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(k2j)}}  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(n1)6a(n2), 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 n = 2 of array A099594.
Cf. A131109, A038573, A001047, A106516.
Sequence in context: A029868 A030267 A026290 * A049221 A081670 A124805
Adjacent sequences: A027646 A027647 A027648 * A027650 A027651 A027652


KEYWORD

nonn,easy,nice,changed


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



