The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A024023 a(n) = 3^n - 1. 87
 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 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 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 Number of terms in conjunctive normal form of Boolean expression with n variables. E.g., a(2) = 8: [~x, ~y, x, y, ~x|~y, ~x|y, x|~y, x|y]. - Yuchun Ji, May 12 2023 Number of rays of the Coxeter arrangement of type B_n. Equivalently, number of facets of the n-dimensional type B permutahedron. - Jose Bastidas, Sep 12 2023 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 and Robert Granger, An efficient deterministic test for Kloosterman sum zeros, Mathematics of Computation, Vol. 83, No. 285 (2014), pp. 347-363; arXiv preprint, arXiv:1104.3882 [math.NT], 2011-2012. See 1st column of Table 2, p. 9. Feryal Alayont and Evan Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4. 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, SIAM J. Discrete Math., Vol. 26, No. 3 (2012), pp. 1104-1107; arXiv preprint, 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, Classical and Quantum Gravity, Vol. 21, No. 22 (2004), pp. 5245--5251; arXiv preprint, arXiv:gr-qc/0407052, 2004. Amir Sapir, The Tower of Hanoi with Forbidden Moves, The Computer J. 47 (1) (2004) 20, case three-in-a row, sequence b(n). Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6. Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT]. Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10. Wikipedia, Herbrand Structure. Damiano Zanardini, Computational Logic, Slides, 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 A128760(a(n)) > 0. - Reinhard Zumkeller, Mar 25 2007 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) = A024101(n)/A034472(n). - Reinhard Zumkeller, Feb 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 Sum_{n>=1} 1/a(n) = A214369. - Amiram Eldar, Nov 11 2020 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: [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). - Reinhard Zumkeller, Mar 11 2010 MATHEMATICA 3^Range[0, 30]-1 (* Paolo Xausa, Jul 15 2023 *) 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) my(x='x+O('x^100)); concat([0], Vec(2*x/(-1+x)/(-1+3*x))) \\ Altug Alkan, Oct 16 2015 CROSSREFS Cf. triangle A013609. Cf. A003462, A007051, A034472, A214369. Cf. second column of A145901. Sequence in context: A124721 A279735 A103453 * A295137 A126966 A002930 Adjacent sequences: A024020 A024021 A024022 * A024024 A024025 A024026 KEYWORD nonn,easy AUTHOR N. J. A. Sloane STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 8 01:04 EDT 2024. Contains 375018 sequences. (Running on oeis4.)