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%I #183 May 18 2024 14:53:15
%S 1,4,14,46,146,454,1394,4246,12866,38854,117074,352246,1058786,
%T 3180454,9549554,28665046,86027906,258149254,774578834,2323998646,
%U 6972520226,20918609254,62757924914,188277969046,564842295746,1694543664454,5083664547794,15251060752246
%N a(n) = 2*(3^n) - 2^n.
%C Poly-Bernoulli numbers B_n^(k) with k=-2.
%C Binomial transform of A007051, if both sequences start at 0. Binomial transform of A000225(n+1). - _Paul Barry_, Mar 24 2003
%C 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
%C 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
%C With regard to the comment by Ross La Haye: For proper subsets see A056182. - For nonempty subsets see A091344. - For nonempty proper subsets see a(n+1) in A260217. - _Manfred Boergens_, Aug 02 2023
%C 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
%C Equals row sums of the triangular version of A038573. - _Gary W. Adamson_, Jun 04 2009
%C Inverse binomial transform of A085350. - _Paul Curtz_, Nov 14 2009
%C 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
%C Row sums of Riordan triangle A106516. - _Wolfdieter Lang_, Jan 09 2015
%C 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
%C 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 n-th position a(n) times out of every 6^n. - _Peter Munn_, May 18 2017
%C Also row sums of A124929. - _Omar E. Pol_, Jun 15 2017
%C 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
%C a(n) is also the number of acyclic orientations of the complete bipartite graph K_{2,n}. - _Vincent Pilaud_, Sep 15 2020
%C a(n-1) is also the number of n-digit numbers whose largest decimal digit is 2. - _Stefano Spezia_, Nov 15 2023
%D Leonhard Euler, Introductio in analysin infinitorum (1748), section 216.
%H T. D. Noe, <a href="/A027649/b027649.txt">Table of n, a(n) for n = 0..200</a>
%H Taylor Brysiewicz, Holger Eble, and Lukas Kühne, <a href="https://arxiv.org/abs/2105.14542">Enumerating chambers of hyperplane arrangements with symmetry</a>, arXiv:2105.14542 [math.CO], 2021.
%H R. E. Crandall, <a href="http://dx.doi.org/10.1090/S0025-5718-1978-0480321-3">On the 3x+1 problem</a>, Math. Comp., 32 (1978) 1281-1292.
%H Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1503.01796">A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata</a>, arXiv:1503.01796 [math.CO], 2015; see also the <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/CAcount.html">Accompanying Maple Package</a>.
%H Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1503.04249">Odd-Rule Cellular Automata on the Square Grid</a>, arXiv:1503.04249 [math.CO], 2015.
%H John Elias, <a href="/A027649/a027649_1.png">Illustration of initial terms: Reflected Sierpinski triangle</a>
%H K. Imatomi, M. Kaneko, and E. Takeda, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Kaneko/kaneko2.html">Multi-Poly-Bernoulli Numbers and Finite Multiple Zeta Values</a>, J. Int. Seq. 17 (2014) # 14.4.5
%H Ken Kamano, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Kamano/kamano5.html">Sums of Products of Poly-Bernoulli Numbers of Negative Index</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.1.3.
%H K. Kamano, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Kamano/kamano2.html">Sums of Products of Bernoulli Numbers, Including Poly-Bernoulli Numbers</a>, J. Int. Seq. 13 (2010), 10.5.2.
%H Masanobu Kaneko, <a href="http://www.numdam.org/item?id=JTNB_1997__9_1_221_0">Poly-Bernoulli numbers</a>, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
%H Takao Komatsu, <a href="https://doi.org/10.22436/jnsa.012.12.05">Some recurrence relations of poly-Cauchy numbers</a>, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
%H Ross La Haye, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/LaHaye/lahaye5.html">Binary Relations on the Power Set of an n-Element Set</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
%H S. Nkonkobe and V. Murali, <a href="http://arxiv.org/abs/1509.07352"> On some properties and relations between restricted barred preferential arrangements, multi-poly-bernoulli numbers and related numbers</a>, arXiv:1509.07352 [math.CO], 2015.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SierpinskiSieve.html">Sierpinski Sieve</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Sierpi%C5%84ski_triangle">Sierpinski triangle</a>
%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6).
%F G.f.: (1-x)/((1-2*x)*(1-3*x)).
%F a(n) = 3*a(n-1) + 2^(n-1), with a(0) = 1.
%F a(n) = Sum_{k=0..n} binomial(n, k)*(2^(k+1) - 1). - _Paul Barry_, Mar 24 2003
%F Partial sums of A053581. - _Paul Barry_, Jun 26 2003
%F 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
%F a(n) = A090888(n, 3). - _Ross La Haye_, Sep 21 2004
%F 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
%F a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(n,j)*binomial(j+1,k+1). - _Paul Barry_, Sep 18 2006
%F a(n) = A166060(n+1)/6. - _Philippe Deléham_, Oct 21 2009
%F a(n) = 5*a(n-1) - 6*a(n-2), a(0)=1, a(1)=4. - _Harvey P. Dale_, Apr 22 2011
%F a(n) = A217764(n,2). - _Ross La Haye_, Mar 27 2013
%F For n>0, a(n) = 3 * a(n-1) + 2^(n-1) = 2 * (a(n-1) + 3^(n-1)). - _J. Conrad_, Oct 29 2015
%F 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
%F E.g.f.: exp(2*x)*(2*exp(x) - 1). - _Stefano Spezia_, May 18 2024
%p a(n, k):= (-1)^n*sum( (-1)^'m'*'m'!*Stirling2(n,'m')/('m'+1)^k,'m'=0..n);
%p seq(a(n, -2), n=0..30);
%t Table[2(3^n)-2^n,{n,0,30}] (* or *) LinearRecurrence[ {5,-6},{1,4},31] (* _Harvey P. Dale_, Apr 22 2011 *)
%o (PARI) a(n)=2*(3^n)-2^n \\ _Charles R Greathouse IV_, Jul 16, 2011
%o (PARI) Vec((1-x)/((1-2*x)*(1-3*x)) + O(x^50)) \\ _Altug Alkan_, Oct 12 2015
%o (Magma) [2*(3^n)-2^n: n in [0..30]]; // _Vincenzo Librandi_, Jul 17 2011
%o (Haskell)
%o a027649 n = a027649_list !! n
%o a027649_list = map fst $ iterate (\(u, v) -> (3 * u + v, 2 * v)) (1, 1)
%o -- _Reinhard Zumkeller_, Jun 09 2013
%o (SageMath) [2*(3^n - 2^(n-1)) for n in (0..30)] # _G. C. Greubel_, Aug 01 2022
%Y Row n = 2 of array A099594.
%Y Also occurs as a row, column, diagonal or as row sums in A038573, A085870, A090888, A106516, A217764, A281891.
%Y Cf. A000079, A000225, A001047, A007051, A053581, A056182, A085350, A091344, A124929, A166060, A260217, A318921.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E Better formulas from _David W. Wilson_ and _Michael Somos_
%E Incorrect formula removed by _Charles R Greathouse IV_, Mar 18 2010
%E Duplications (due to corrections to A numbers) removed by _Peter Munn_, Jun 15 2017
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