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0, 2, 8, 26, 80, 242, 728, 2186, 6560, 19682, 59048, 177146, 531440, 1594322, 4782968, 14348906, 43046720, 129140162, 387420488, 1162261466, 3486784400, 10460353202, 31381059608, 94143178826, 282429536480, 847288609442
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Number of different directions along lines and hyper-diagonals in an n-dimensional cubic lattice for the attacking queens problem (A036464 in n=2, A068940 in n=3 and A068941 in n=4). The n-dimensional direction vectors have the a(n)+1 Cartesian coordinates (i,j,k,l,..) where i,j,k,l..=-1,0 or +1, excluding the zero-vector i=j=k=l=..=0. The corresponding hyper-line count is A003462. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 01 2006
Numbers n for which the expression 3^n/(n+1) is an integer. - Paolo P. Lava (paoloplava(AT)gmail.com), May 29 2006
A128760(a(n)) > 0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 25 2007
Total number of sequences of length m=1,...,n with nonzero integer elements satisfying the condition sum(|n_k|,k=1..m) <= n. See the K. A. Meissner example in arXiv:gr-qc/0407052v1, p. 6 (with a typo: it should be 3^([2a]-1)-1). W. Lang, Jan 21 2008.
Draisma et al. prove that the number of lattice directions in which an n-dimensional convex body in R^n has minimal width is at most 3^n-1, with equality only for the regular cross-polytope, sharpening the 3^d-theorem of Hermann Minkowski. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 13 2009]
a(n) = A24101(n)/A034472(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 14 2009]
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if x and y are disjoint and either 0) x is a proper subset of y or y is a proper subset of x, or 1) x is not a subset of y and y is not a subset of x. Then a(n) = |R|. [From Ross La Haye (rlahaye(AT)new.rr.com), Mar 19 2009]
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REFERENCES
| 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), Mar 19 2009]
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for sequences related to linear recurrences with constant coefficients, signature (4,-3).
Jan Draisma, Tyrrell B. McAllister and Benjamin Nill, Lattice width directions and Minkowski's 3^d-theorem, Jan 10, 2009. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 13 2009]
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FORMULA
| a(n)=2*A003462(n) - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 01 2006
G.f.: 2*x/(-1+x)/(-1+3*x) = 1/(-1+x)-1/(-1+3*x). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 19 2007
a(n)=sum(sum(binomial(k-1,m-1)*2^m,m=1..k),k=1..n), n>=1. a(0)=0. From the sequence combinatorics mentioned above. Twice partial sums of powers of 3.
E.g.f.: e^(3*x)-e^x. [From Mohammad K. Azarian (azarian(AT)evansville.edu), Jan 14 2009]
a(n) = 3*a(n-1)+2 (with a(0)=0). [From Vincenzo Librandi, Nov 19 2010]
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EXAMPLE
| Ternary......decimal (comment from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 14 2007):
0...............0
2...............2
22..............8
222............26
2222...........80
22222.........242
222222........728
2222222......2186
22222222.....6560
222222222...19682
2222222222..59048
etc...........etc.
Sequence combinatorics: n=3: With length m=1: [1],[2],[3] each with 2 signs, with m=2: [1,1], [1,2], [2,1], each 2^2=4 times from choosing signs; m=3: [1,1,1] coming in 2^3 signed versions: 3*2+3*4+1*8= 26 = a(3). The order is important, hence the M_0 multinomials A048996 enter as factors.
A027902 gives the 384 divisors of a(24). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 11 2010]
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MAPLE
| g:=1/(1-3*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)-1, n=0..31); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 09 2009]
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MATHEMATICA
| a=2; lst={0, a}; Do[a=a*3+2; AppendTo[lst, a], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 25 2008]
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PROG
| (MAGMA) [3^n-1: n in [0..35]]; // Vincenzo Librandi, Apr 30 2011
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CROSSREFS
| Cf. triangle A013609.
Cf. A003462, A007051, A034472 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 25 2008]
Sequence in context: A097040 A124721 * A103453 A126966 A002930 A060410
Adjacent sequences: A024020 A024021 A024022 * A024024 A024025 A024026
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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