A006101 -id:A006101 - cp's OEIS frontend

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

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

Offset: 0

N. J. A. Sloane

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

			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

Mordechai Ben-Ari, Mathematical Logic for Computer Science, Third edition, 173-203.

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.
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).

Cf. triangle A013609.
Cf. A003462, A007051, A034472, A214369.
Cf. second column of A145901.

a024023 = subtract 1 . a000244  -- Reinhard Zumkeller, Jun 30 2013

[3^n-1: n in [0..35]]; // Vincenzo Librandi, Apr 30 2011

3^Range[0,30]-1 (* Paolo Xausa, Jul 15 2023 *)

a(n)=3^n-1 \\ Charles R Greathouse IV, Sep 24 2015

vector(50, n, sum(k=0, n, 2^k*binomial(n-1, k))-1) \\ Altug Alkan, Oct 04 2015

my(x='x+O('x^100)); concat([0], Vec(2*x/(-1+x)/(-1+3*x))) \\ Altug Alkan, Oct 16 2015

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
a(n) = Sum_{k=1..n} 2^k*binomial(n,k). - Ridouane Oudra, Jun 15 2025
From Peter Bala, Jul 01 2025: (Start)
For n >= 1, a(2*n)/a(n) = A034472(n) and a(3*n)/a(n) = A034513(n).
Modulo differences in offsets, exp( Sum_{n >= 1} a(k*n)/a(n)*x^n/n ) is the o.g.f. of A003462 (k = 2), A006100 (k = 3), A006101 (k = 4), A006102 (k = 5), A022196 (k = 6), A022197 (k = 7), A022198 (k = 8), A022199 (k = 9), A022200 (k = 10), A022201 (k = 11), A022202 (k = 12) and A022203 (k = 13).
The following are all examples of telescoping series:
Sum_{n >= 1} 3^n/(a(n)*a(n+1)) = 1/2^2; Sum_{n >= 1} 3^n/(a(n)*a(n+1)*a(n+2)) = 1/(2*8^2).
In general, for k >= 1, Sum_{n >= 1} 3^n/(a(n)*a(n+1)*...*a(n+k)) = 1/(a(1)*a(2)*...*a(k)*a(k)).
Sum_{n >= 1} 3^n/(a(n)*a(n+2)) = 5/64; Sum_{n >= 1} (-3)^n/(a(n)*a(n+2)) = -3/64.
Sum_{n >= 1} 3^n/(a(n)*a(n+4)) = 703/83200; Sum_{n >= 1} (-3)^n/(a(n)*a(n+4)) = - 417/83200. (End)

A022167 Triangle of Gaussian binomial coefficients [ n,k ] for q = 3.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 13, 13, 1, 1, 40, 130, 40, 1, 1, 121, 1210, 1210, 121, 1, 1, 364, 11011, 33880, 11011, 364, 1, 1, 1093, 99463, 925771, 925771, 99463, 1093, 1, 1, 3280, 896260, 25095280, 75913222, 25095280, 896260, 3280, 1

Offset: 0

N. J. A. Sloane

The coefficients of the matrix inverse are apparently given by T^(-1)(n,k) = (-1)^n*A157783(n,k). - R. J. Mathar, Mar 12 2013

			Triangle begins:
  1;
  1,    1;
  1,    4,      1;
  1,   13,     13,        1;
  1,   40,    130,       40,        1;
  1,  121,   1210,     1210,      121,        1;
  1,  364,  11011,    33880,    11011,      364,      1;
  1, 1093,  99463,   925771,   925771,    99463,   1093,    1;
  1, 3280, 896260, 25095280, 75913222, 25095280, 896260, 3280, 1;

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

T. D. Noe, Rows n = 0..50 of triangle, flattened
R. Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
Index entries for sequences related to Gaussian binomial coefficients

Columns k=0..3 give A000012, A003462, A006100, A006101.

Cf. A006117 (row sums).

A022167 := proc(n,m)
        A027871(n)/A027871(n-m)/A027871(m) ;
end proc:
seq(seq(A022167(n,m),m=0..n),n=0..10) ; # R. J. Mathar, Nov 14 2011

p[n_] := Product[3^k-1, {k, 1, n}]; t[n_, m_] := p[n]/(p[n-m]*p[m]); Table[t[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jan 10 2014, after R. J. Mathar *)
Table[QBinomial[n, k, 3], {n, 0, 10}, {k, 0, n}] // Flatten
(* or, after Vladimir Kruchinin, using S for qStirling2: *)
S[n_, k_, q_] /; 1 <= k <= n := S[n-1, k-1, q] + Sum[q^j, {j, 0, k-1}]* S[n-1, k, q]; S[n_, 0, ] := KroneckerDelta[n, 0]; S[0, k, ] := KroneckerDelta[0, k]; S[, , ] = 0;
T[n_, k_] /; n >= k := Sum[Binomial[n, j]*S[n-j, n-k, q]*(q-1)^(k-j) /. q -> 3, {j, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 08 2020 *)

T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - Peter A. Lawrence, Jul 13 2017
T(n,k) = Sum_{j=0..k} C(n,j)*qStirling2(n-j,n-k,3)*(2)^(k-j),j,0,k), n >= k, where qStirling2(n,k,3) is triangle A333143. - Vladimir Kruchinin, Mar 07 2020
G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 3^j - 1. - Seiichi Manyama, May 09 2025

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

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A022167 Triangle of Gaussian binomial coefficients [ n,k ] for q = 3.

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A383754 Expansion of 1/Product_{k=0..3} (1 - 2^k * 3^(3-k) * x).

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