

A027649


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


37



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
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(2*(3^n)  2^n)). 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
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
This is also row 2 of A281891: for n >= 1, when consecutive positive integers are written as a product of primes in nondecreasing order, a factor of 2 or 3 occurs in nth position a(n) times out of every 6^n.  Peter Munn, May 18 2017
Also row sums of A124929.  Omar E. Pol, Jun 15 2017
This is the sum of A318921(n) for n in the range 2^(k+1) to 2^(k+2)1. See A318921 for proof.  N. J. A. Sloane, Sep 25 2018
a(n) is also the number of acyclic orientations of the complete bipartite graph K_{2,n}.  Vincent Pilaud, Sep 15 2020


REFERENCES

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


LINKS

T. D. Noe, Table of n, a(n) for n = 0..200
Taylor Brysiewicz, Holger Eble, and Lukas Kühne, Enumerating chambers of hyperplane arrangements with symmetry, arXiv:2105.14542 [math.CO], 2021.
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 [math.CO], 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 [math.CO], 2015.
John Elias, Illustration of initial terms: Reflected Sierpinski triangle
K. Imatomi, M. Kaneko, and E. Takeda, MultiPolyBernoulli Numbers and Finite Multiple Zeta Values, J. Int. Seq. 17 (2014) # 14.4.5
Ken Kamano, Sums of Products of PolyBernoulli Numbers of Negative Index, Journal of Integer Sequences, Vol. 15 (2012), #12.1.3.
K. Kamano, Sums of Products of Bernoulli Numbers, Including PolyBernoulli Numbers, J. Int. Seq. 13 (2010), 10.5.2.
Masanobu Kaneko, PolyBernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221228.
Takao Komatsu, Some recurrence relations of polyCauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829845.
Ross La Haye, Binary Relations on the Power Set of an nElement 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, multipolybernoulli numbers and related numbers, arXiv:1509.07352 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Sierpinski Sieve
Wikipedia, Sierpinski triangle
Index entries for sequences related to Bernoulli numbers.
Index entries for linear recurrences with constant coefficients, signature (5,6).


FORMULA

G.f.: (1x)/((12*x)*(13*x)).
a(n) = 3*a(n1) + 2^(n1), with a(0) = 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(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} binomial(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} binomial(n,j)*binomial(j+1,k+1).  Paul Barry, Sep 18 2006
a(n) = A166060(n+1)/6.  Philippe Deléham, Oct 21 2009
a(n) = 5*a(n1)  6*a(n2), 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(n1) + 2^(n1) = 2 * (a(n1) + 3^(n1)).  J. Conrad, Oct 29 2015
for n>0, a(n) = 2 * (1 + 2^(n2) + Sum_{x=1..n2} Sum_{k=0..x1} (binomial(x1,k)*(2^(k+1) + 2^(nx+k)))).  J. Conrad, Dec 10 2015


MAPLE

a(n, k):= (1)^n*sum( (1)^'m'*'m'!*Stirling2(n, 'm')/('m'+1)^k, 'm'=0..n);
seq(a(n, 2), n=0..30);


MATHEMATICA

Table[2(3^n)2^n, {n, 0, 30}] (* or *) LinearRecurrence[ {5, 6}, {1, 4}, 31] (* Harvey P. Dale, Apr 22 2011 *)


PROG

(PARI) a(n)=2*(3^n)2^n \\ Charles R Greathouse IV, Jul 16, 2011
(PARI) Vec((1x)/((12*x)*(13*x)) + O(x^50)) \\ Altug Alkan, Oct 12 2015
(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
(SageMath) [2*(3^n  2^(n1)) for n in (0..30)] # G. C. Greubel, Aug 01 2022


CROSSREFS

Row n = 2 of array A099594.
Also occurs as a row, column, diagonal or as row sums in A038573, A085870, A090888, A106516, A217764, A281891.
Cf. A000079, A000225, A001047, A053581, A085350, A166060, A318921.
Sequence in context: A029868 A030267 A026290 * A330796 A049221 A081670
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
Duplications (due to corrections to A numbers) removed by Peter Munn, Jun 15 2017


STATUS

approved



