The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS"). Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A024023 a(n) = 3^n - 1. 78
 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 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 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. 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. 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) = 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: ,, 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, A214369. Sequence in context: A124721 A279735 A103453 * A295137 A126966 A002930 Adjacent sequences:  A024020 A024021 A024022 * A024024 A024025 A024026 KEYWORD nonn,easy,changed AUTHOR STATUS approved

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

Last modified December 1 00:44 EST 2020. Contains 338831 sequences. (Running on oeis4.)