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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; internal format)
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OFFSET
| 0,2
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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 (pbarry(AT)wit.ie), 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 (rlahaye(AT)new.rr.com), 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 (thedavinci(AT)gmail.com), Feb 15 2009]
Equals row sums of the triangular version of A038573. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 04 2009]
Inverse binomial transform of A085350. [From Paul Curtz (bpcrtz(AT)free.fr), 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]
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REFERENCES
| R. E. Crandall, On the "3x+1" Problem, Math. Comp. 32 (1978), 1281-1292.
Leonhard Euler, Introduction in analysin infinitorum (1748), section 216.
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. [From Ross La Haye (rlahaye(AT)new.rr.com), Feb 22 2009]
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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).
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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 (pbarry(AT)wit.ie), Mar 24 2003
Partial sums of A053581. - Paul Barry (pbarry(AT)wit.ie), 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 (benoit7848c(AT)orange.fr), Aug 05 2003
a(n) = A090888(n, 3). - Ross La Haye (rlahaye(AT)new.rr.com), 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 (pbarry(AT)wit.ie), Apr 17 2005
a(n)=sum{k=0..n, sum{j=0..n, C(n,j)C(j+1,k+1)}}; - Paul Barry (pbarry(AT)wit.ie), Sep 18 2006
Row sums of triangle A131109. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 15 2007
a(n)= A166060(n+1)/6. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 21 2009]
a(n) = 5a(n-1)-6a(n-2), a(0)=1, a(1)=4. [From Harvey P. Dale, Apr 22 2011]
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MAPLE
| (-1)^n*sum( (-1)^'m'*'m'!*stirling2(n, 'm')/('m'+1)^k, 'm'=0..n);
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MATHEMATICA
| Table[2(3^n)-2^n, {n, 0, 25}] (* or *)
LinearRecurrence[{5, -6}, {1, 4}, 25] (* From Harvey P. Dale, Apr 22 2011 *)
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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
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CROSSREFS
| Row 2 of array A099594.
Cf. A131109.
A038573 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 04 2009]
Sequence in context: A029868 A030267 A026290 * A049221 A081670 A124805
Adjacent sequences: A027646 A027647 A027648 * A027650 A027651 A027652
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Better formulae from David W. Wilson (davidwwilson(AT)comcast.net) and Michael Somos.
Incorrect formula removed by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Mar 18 2010
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