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 A024023 a(n) = 3^n - 1. 70
 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, 2541865828328, 7625597484986, 22876792454960 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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, May 01 2006 Numbers n for which the expression 3^n/(n+1) is an integer. - Paolo P. Lava, May 29 2006 A128760(a(n)) > 0. - Reinhard Zumkeller, Mar 25 2007 Total number of sequences of length m=1,...,n with nonzero integer elements satisfying the condition Sum_{k=1..m} |n_k| <= n. See the K. A. Meissner link p. 6 (with a typo: it should be 3^([2a]-1)-1). - Wolfdieter 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. - Jonathan Vos Post, Jan 13 2009 [reworded from the Draisma et al. reference, Joerg Arndt, Dec 31 2017] a(n) = A024101(n)/A034472(n). - Reinhard Zumkeller, 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|. - Ross La Haye, Mar 19 2009 Number of neighbors in Moore's neighborhood in n dimensions. - Dmitry Zaitsev, Nov 30 2015 REFERENCES Mordechai Ben-Ari, Mathematical Logic for Computer Science, Third edition, 173-203. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Omran Ahmadi, Robert Granger, An efficient deterministic test for Kloosterman sum zeros, arXiv:1104.3882 [math.NT], 2011-2012. See 1st column of Table 2 p. 9. Michael Baake, Franz Gähler, and Uwe Grimm, Examples of Substitution Systems and Their Factors, Journal of Integer Sequences, Vol. 16 (2013), #13.2.14. R. Samuel Buss, Herbrand's Theorem, University of California, Logic and Computational Complexity pp. 195-209, Lecture Notes in Computer Science, vol 960. Springer. Jan Draisma, Tyrrell B. McAllister and Benjamin Nill, Lattice width directions and Minkowski's 3^d-theorem, arXiv:0901.1375 [math.CO], Jan 10 2009. - Jonathan Vos Post, Jan 13 2009 Alessandro Farinelli, Herbrand Universe and Herbrand Base 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. Krzysztof A. Meissner, Black hole entropy in Loop Quantum Gravity, arXiv:gr-qc/0407052, 2004. Wikipedia, Herbrand Structure Damiano Zanardini, Computational Logic, UPM European Master in Computational Logic (EMCL) School of Computer Science Technical University of Madrid, 2009-2010. Index entries for linear recurrences with constant coefficients, signature (4,-3). FORMULA a(n) = A000244(n) - 1. a(n) = 2*A003462(n). - R. J. Mathar, May 01 2006 G.f.: 2*x/(-1+x)/(-1+3*x) = 1/(-1+x)-1/(-1+3*x). - R. J. Mathar, Nov 19 2007 a(n) = Sum_{k=1..n} Sum_{m=1..k} binomial(k-1,m-1)*2^m, 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. - Mohammad K. Azarian, Jan 14 2009 a(n) = 3*a(n-1) + 2 (with a(0)=0). - Vincenzo Librandi, Nov 19 2010 E.g.f.: -E(0) where E(k) = 1 - 3^k/(1 - x/(x - 3^k*(k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2012 a(n) = A227048(n,A020914(n)). - Reinhard Zumkeller, Jun 30 2013 EXAMPLE From Zerinvary Lajos, Jan 14 2007: (Start) Ternary......decimal: 0...............0 2...............2 22..............8 222............26 2222...........80 22222.........242 222222........728 2222222......2186 22222222.....6560 222222222...19682 2222222222..59048 etc...........etc. (End) Sequence combinatorics: n=3: With length m=1: ,, 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). - Reinhard Zumkeller, Mar 11 2010 MAPLE g:=1/(1-3*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)-1, n=0..31); # Zerinvary Lajos, Jan 09 2009 MATHEMATICA a := 0; a[n_] := a[n - 1] + 2*3^(n - 1) (* Fred Daniel Kline, Feb 09 2014 *) PROG (MAGMA) [3^n-1: n in [0..35]]; // Vincenzo Librandi, Apr 30 2011 (Haskell) a024023 = subtract 1 . a000244  -- Reinhard Zumkeller, Jun 30 2013 (PARI) a(n)=3^n-1 \\ Charles R Greathouse IV, Sep 24 2015 (PARI) vector(50, n, sum(k=0, n, 2^k*binomial(n-1, k))-1) \\ Altug Alkan, Oct 04 2015 (PARI) x='x+O('x^100); concat(, Vec(2*x/(-1+x)/(-1+3*x))) \\ Altug Alkan, Oct 16 2015 CROSSREFS Cf. triangle A013609. Cf. A003462, A007051, A034472. Sequence in context: A124721 A279735 A103453 * A295137 A126966 A002930 Adjacent sequences:  A024020 A024021 A024022 * A024024 A024025 A024026 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified October 20 02:03 EDT 2019. Contains 328244 sequences. (Running on oeis4.)