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A024023 a(n) = 3^n - 1. 77
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.

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, 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: [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

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] := 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([0], 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

N. J. A. Sloane

STATUS

approved

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Last modified September 18 18:41 EDT 2020. Contains 337172 sequences. (Running on oeis4.)