A000244 -id:A000244 - cp's OEIS frontend

A002379 a(n) = floor(3^n / 2^n).

1, 1, 2, 3, 5, 7, 11, 17, 25, 38, 57, 86, 129, 194, 291, 437, 656, 985, 1477, 2216, 3325, 4987, 7481, 11222, 16834, 25251, 37876, 56815, 85222, 127834, 191751, 287626, 431439, 647159, 970739, 1456109, 2184164, 3276246, 4914369, 7371554, 11057332

Offset: 0

N. J. A. Sloane

It is an important unsolved problem related to Waring's problem to show that a(n) = floor((3^n-1)/(2^n-1)) holds for all n > 1. This has been checked for 10000 terms and is true for all sufficiently large n, by a theorem of Mahler. [Lichiardopol]
a(n) = floor((3^n-1)/(2^n-1)) holds true at least for 2 <= n <= 305000. - Hieronymus Fischer, Dec 31 2008
a(n) is also the curve length (rounded down) of the Sierpiński arrowhead curve after n iterations, let a(0) = 1. - Kival Ngaokrajang, May 21 2014
a(n) is composite infinitely often (Forman and Shapiro). More exactly, a(n) is divisible by at least one of 2, 5, 7 or 11 infinitely often (Dubickas and Novikas). - Tomohiro Yamada, Apr 15 2017

R. K. Guy, Unsolved Problems in Number Theory, E19.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 82.
S. S. Pillai, On Waring's problem, J. Indian Math. Soc., 2 (1936), 16-44.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Arturas Dubickas and Aivaras Novikas, Integer parts of powers of rational numbers, Math. Z. 251 (2005), 635--648, available from the first author's page.
W. Forman and H. N. Shapiro, An arithmetic property of certain rational powers, Comm. Pure. Appl. Math. 20 (1967), 561-573.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
N. Lichiardopol, Problem 925 (BCC20.19), A number-theoretic problem, in Research Problems from the 20th British Combinatorial Conference, Discrete Math., 308 (2008), 621-630.
K. Mahler, On the fractional parts of the powers of a rational number, II, Mathematika 4 (1957), 122-124.
Kival Ngaokrajang, Illustration of Sierpinski arrowhead curve for n = 0..5
Eric Weisstein's World of Mathematics, Power Floors
Wikipedia, Sierpiński arrowhead curve

Cf. A094969-A094500, A000217, A081464, A153662, A153665, A153666.
Cf. A060692, A002380, A000079, A000244.
Cf. A046037, A070758, A070759, A067904 (Composites and Primes).
Cf. A064628 (an analog for 4/3).

a002379 n = 3^n `div` 2^n  -- Reinhard Zumkeller, Jul 11 2014

[Floor(3^n / 2^n): n in [0..40]]; // Vincenzo Librandi, Sep 08 2011

A002379:=n->floor(3^n/2^n); seq(A002379(k), k=0..100); # Wesley Ivan Hurt, Oct 29 2013

Table[Floor[(3/2)^n], {n, 0, 40}] (* Robert G. Wilson v, May 11 2004 *)
x[n_] := -(1/2) + (3/2)^n + ArcTan[Cot[(3/2)^n Pi]]/Pi; Array[x, 40] (* Fred Daniel Kline, Dec 21 2017 *)
x[n_]:=Round[-(1/2) + (3^n - 1)/(2^n - 1)]; Array[x, 39, 2] (* offset n+1, Fred Daniel Kline, Apr 13 2018 *)

makelist(floor(3^n/2^n), n, 0, 50); /* Martin Ettl, Oct 17 2012 */

a(n)=3^n>>n \\ Charles R Greathouse IV, Jun 10 2011

def A002379(n): return 3**n>>n # Chai Wah Wu, Sep 21 2022

a(n) = b(n) - (-2/3)^n where b(n) is defined by the recursion b(0):=2, b(1):=5/6, b(n+1):=(5/6)*b(n) + b(n-1). - Hieronymus Fischer, Dec 31 2008
a(n) = (1/2)*(b(n) + sqrt(b(n)^2 - (-4)^n)) (with b(n) as defined above). - Hieronymus Fischer, Dec 31 2008
3^n = a(n)*2^n + A002380(n). - R. J. Mathar, Oct 26 2012
a(n) = -(1/2) + (3/2)^n + arctan(cot((3/2)^n Pi)) / Pi. - Fred Daniel Kline, Apr 14 2018
a(n+1) = round( -(1/2) + (3^n-1)/(2^n-1) ). - Fred Daniel Kline, Apr 14 2018

More terms from Robert G. Wilson v, May 11 2004

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

Original entry on oeis.org

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)

A025192 a(0)=1; a(n) = 2*3^(n-1) for n >= 1.

Original entry on oeis.org

1, 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, 1062882, 3188646, 9565938, 28697814, 86093442, 258280326, 774840978, 2324522934, 6973568802, 20920706406, 62762119218, 188286357654, 564859072962, 1694577218886, 5083731656658, 15251194969974

Offset: 0

Clark Kimberling

Warning: there is considerable overlap between this entry and the essentially identical A008776.
Shifts one place left when plus-convolved (PLUSCONV) with itself. a(n) = 2*Sum_{i=0..n-1} a(i). - Antti Karttunen, May 15 2001
Let M = { 0, 1, ..., 2^n-1 } be the set of all n-bit numbers. Consider two operations on this set: "sum modulo 2^n" (+) and "bitwise exclusive or" (XOR). The results of these operations are correlated.
To give a numerical measure, consider the equations over M: u = x + y, v = x XOR y and ask for how many pairs (u,v) is there a solution? The answer is exactly a(n) = 2*3^(n-1) for n >= 1. The fraction a(n)/4^n of such pairs vanishes as n goes to infinity. - Max Alekseyev, Feb 26 2003
Number of (s(0), s(1), ..., s(2n+2)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+2, s(0) = 3, s(2n+2) = 3. - Herbert Kociemba, Jun 10 2004
Number of compositions of n into parts of two kinds. For a string of n objects, before the first, choose first kind or second kind; before each subsequent object, choose continue, first kind, or second kind. For example, compositions of 3 are 3; 2,1; 1,2; and 1,1,1. Using parts of two kinds, these produce respectively 2, 4, 4 and 8 compositions, 2+4+4+8 = 18. - Franklin T. Adams-Watters, Aug 18 2006
In the compositions the kinds of parts are ordered inside a run of identical parts, see example.  Replacing "ordered" by "unordered" gives A052945. - Joerg Arndt, Apr 28 2013
Number of permutations of {1, 2, ..., n+1} such that no term is more than 2 larger than its predecessor. For example, a(3) = 18 because all permutations of {1, 2, 3, 4} are valid except 1423, 1432, 2143, 3142, 2314, 3214, in which 1 is followed by 4. Proof: removing (n + 1) gives a still-valid sequence. For n >= 2, can insert (n + 1) either at the beginning or immediately following n or immediately following (n - 1), but nowhere else. Thus the number of such permutations triples when we increase the sequence length by 1. - Joel B. Lewis, Nov 14 2006
Antidiagonal sums of square array A081277. - Philippe Deléham, Dec 04 2006
Equals row sums of triangle A160760. - Gary W. Adamson, May 25 2009
Let M = a triangle with (1, 2, 4, 8, ...) as the left border and all other columns = (0, 1, 2, 4, 8, ...). A025192 = lim_{n->oo} M^n, the left-shifted vector considered as a sequence. - Gary W. Adamson, Jul 27 2010
Number of nonisomorphic graded posets with 0 and uniform hasse graph of rank n with no 3-element antichain. ("Uniform" used in the sense of Retakh, Serconek and Wilson. By "graded" we mean that all maximal chains have the same length n.) - David Nacin, Feb 13 2012
Equals partial sums of A003946 prefaced with a 1: (1, 1, 4, 12, 36, 108, ...). - Gary W. Adamson, Feb 15 2012
Number of vertices (or sides) of the (n-1)-th iteration of a Gosper island. - Arkadiusz Wesolowski, Feb 07 2013
Row sums of triangle in A035002. - Jon Perry, May 30 2013
a(n) counts walks (closed) on the graph G(1-vertex; 1-loop, 1-loop, 2-loop, 2-loop, 3-loop, 3-loop, ...). - David Neil McGrath, Jan 01 2015
From Tom Copeland, Dec 03 2015: (Start)
For n > 0, a(n) are the traces of the even powers of the adjacency matrix M of the simple Lie algebra B_3, tr(M^(2n)) where M = Matrix(row 1; row 2; row 3) = Matrix[0,1,0; 1,0,2; 0,1,0], same as the traces of Matrix[0,2,0; 1,0,1; 0,1,0] (cf. Damianou). The traces of the odd powers vanish.
The characteristic polynomial of M equals determinant(x*I - M) = x^3 - 3x = A127672(3,x), so 1 - 3*x^2 = det(I - x M) = exp(-Sum_{n>=1} tr(M^n) x^n / n), implying Sum_{n>=1} a(n+1) x^(2n) / (2n) = -log(1 - 3*x^2), giving a logarithmic generating function for the aerated sequence, excluding a(0) and a(1).
a(n+1) = tr(M^(2n)), where tr(M^n) = 3^(n/2) + (-1)^n * 3^(n/2) = 2^n*(cos(Pi/6)^n + cos(5*Pi/6)^n) = n-th power sum of the eigenvalues of M = n-th power sum of the zeros of the characteristic polynomial.
The relation det(I - x M) = exp(-Sum_{n>=1} tr(M^n) x^n / n) = Sum_{n>=0} P_n(-tr(M), -tr(M^2), ..., -tr(M^n)) x^n/n! = exp(P.(-tr(M), -tr(M^2), ...)x), where P_n(x(1), ..., x(n)) are the partition polynomials of A036039 implies that with x(2n) = -tr(M^(2n)) = -a(n+1) for n > 0 and x(n) = 0 otherwise, the partition polynomials evaluate to zero except for P_2(x(1), x(2)) = P_2(0,-6) = -6.
Because of the inverse relation between the partition polynomials of A036039 and the Faber polynomials F_k(b1,b2,...,bk) of A263916, F_k(0,-3,0,0,...) = tr(M^k) gives aerated a(n), excluding n=0,1. E.g., F_2(0,-3) = -2(-3) = 6, F_4(0,-3,0,0) = 2 (-3)^2 = 18, and F_6(0,-3,0,0,0,0) = -2(-3)^3 = 54. (Cf. A265185.)
(End)
Number of permutations of length n > 0 avoiding the partially ordered pattern (POP) {1>2, 1>3, 1>4} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is the largest. - Sergey Kitaev, Dec 08 2020
For n > 0, a(n) is the number of 3-colorings of the grid graph P_2 X P_(n-1). More generally, for q > 1, the number of q-colorings of the grid graph P_2 X P_n is given by q*(q - 1)*((q - 1)*(q - 2) + 1)^(n - 1). - Sela Fried, Sep 25 2023
For n > 1, a(n) is the largest solution to the equation phi(x) = a(n-1). - M. Farrokhi D. G., Oct 25 2023
Number of dotted compositions of degree n. - Diego Arcis, Feb 01 2024

			There are a(3)=18 compositions of 3 into 2 kinds of parts. Here p:s stands for "part p of sort s":
01:  [ 1:0  1:0  1:0  ]
02:  [ 1:0  1:0  1:1  ]
03:  [ 1:0  1:1  1:0  ]
04:  [ 1:0  1:1  1:1  ]
05:  [ 1:0  2:0  ]
06:  [ 1:0  2:1  ]
07:  [ 1:1  1:0  1:0  ]
08:  [ 1:1  1:0  1:1  ]
09:  [ 1:1  1:1  1:0  ]
10:  [ 1:1  1:1  1:1  ]
11:  [ 1:1  2:0  ]
12:  [ 1:1  2:1  ]
13:  [ 2:0  1:0  ]
14:  [ 2:0  1:1  ]
15:  [ 2:1  1:0  ]
16:  [ 2:1  1:1  ]
17:  [ 3:0  ]
18:  [ 3:1  ]
- _Joerg Arndt_, Apr 28 2013
G.f. = 1 + 2*x + 6*x^2 + 18*x^3 + 54*x^4 + 162*x^5 + 486*x^6 + 1458*x^7 + ...

Richard P. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

T. D. Noe, Table of n, a(n) for n = 0..200
George E. Andrews and Greg Simay, Parity palindrome compositions, Integers 21, Paper No A85, (2021), 13 pp.
Diego Arcis, Camilo González, and Sebastián Márquez On the Hopf algebra of noncommutative symmetric functions in superspace, arXiv:2205.11813 [math.CO], 2022.
D. Bevan, D. Levin, P. Nugent, J. Pantone and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510.08036 [math.CO], 2015.
Fan Chung and R. L. Graham, Primitive juggling sequences, Am. Math. Monthly 115 (3) (2008) 185-194.
Pantelis A. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 13.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology, arXiv:1010.6295 [math.RA], 2010-2011.
Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.
Richard P. Stanley, An Equivalence Relation on the Symmetric Group and Multiplicity-free Flag h-Vectors.
Kai Ting Keshia Yap, David Wehlau, and Imed Zaguia, Permutations Avoiding Certain Partially-ordered Patterns, arXiv:2101.12061 [math.CO], 2021.
Vincent Vatter, Counting parity palindrome compositions, arXiv:2109.13155 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (3).

First differences of 3^n (A000244). Other self-convolved sequences: A000108, A007460, A007461, A007462, A007463, A007464, A061922.
Apart from initial term, same as A008776.
Cf. A132019, A160760, A003946, A035002.
Cf. A036039, A263916, A265185, A127672, A328778.

a025192 0 = 1
a025192 n = 2 * 3 ^ (n -1)
a025192_list = 1 : iterate (* 3) 2  -- Reinhard Zumkeller, Nov 27 2012

A025192 := proc(n): if n=0 then 1 else 2*3^(n-1) fi: end: seq(A025192(n),n=0..26);

Join[{1},2*3^(Range[30]-1)]  (* Harvey P. Dale, Mar 22 2011 *)

a(n)=max(1,2*3^(n-1)) \\ Charles R Greathouse IV, Jul 25 2011

Vec((1-x)/(1-3*x) + O(x^100)) \\ Altug Alkan, Dec 05 2015

[1]+[2*3**(n-1) for n in range(1,30)] # David Nacin, Mar 04 2012

G.f.: (1-x)/(1-3*x).
E.g.f.: (2*exp(3*x) + exp(0))/3. - Paul Barry, Apr 20 2003
a(n) = phi(3^n) = A000010(A000244(n)). - Labos Elemer, Apr 14 2003
a(0) = 1, a(n) = Sum_{k=0..n-1} (a(k) + a(n-k-1)). - Benoit Cloitre, Jun 24 2003
a(n) = A002326((3^n-1)/2). - Vladimir Shevelev, May 26 2008
a(1) = 2, a(n) = 3*a(n-1). - Vincenzo Librandi, Jan 01 2011
a(n) = lcm(a(n-1), Sum_{k=1..n-1} a(k)) for n >= 3. - David W. Wilson, Sep 27 2011
a(n) = ((2*n-1)*a(n-1) + (3*n-6)*a(n-2))/(n-1); a(0)=1, a(1)=2. - Sergei N. Gladkovskii, Jul 16 2012
From Sergei N. Gladkovskii, Jul 17 2012: (Start)
For the e.g.f. E(x) = (2/3)*exp(3*x) + exp(0)/3 we have
E(x) = 2*G(0)/3 where G(k) = 1 + k!/(3*(9*x)^k - 3*(9*x)^(2*k+1)/((9*x)^(k+1) + (k+1)!/G(k+1))); (continued fraction, 3rd kind, 3-step).
E(x) = 1+2*x/(G(0)-3*x) where G(k) = 3*x + 1 + k - 3*x*(k+1)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). (End)
a(n) = A114283(0,0). - Reinhard Zumkeller, Nov 27 2012
G.f.: 1 + ((1/2)/G(0) - 1)/x where G(k) = 1 - 2^k/(2 - 4*x/(2*x - 2^k/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 22 2012
G.f.: 1 + x*W(0), where W(k) = 1 + 1/(1 - x*(2*k+3)/(x*(2*k+4) + 1/W(k+1))); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013
G.f.: 1 / (1 - 2*x / (1 - x)). - Michael Somos, Apr 03 2014
Construct the power matrix T(n,j) = [A(n)^*j]*[S(n)^*(j-1)] where A(n)=(2,2,2,...) and S(n)=(0,1,0,0,...). (* is convolution operation.) Then a(n) = Sum_{j=1..n} T(n,j). - David Neil McGrath, Jan 01 2015
G.f.: 1 + 2*x/(1 + 2*x)*( 1 + 5*x/(1 + 5*x)*( 1 + 8*x/(1 + 8*x)*( 1 + 11*x/(1 + 11*x)*( 1 + .... - Peter Bala, May 27 2017
Sum_{n>=0} 1/a(n) = 7/4. - Bernard Schott, Oct 02 2021
From Amiram Eldar, May 08 2023: (Start)
Sum_{n>=0} (-1)^n/a(n) = 5/8.
Product_{n>=1} (1 - 1/a(n)) = A132019. (End)

Additional comments from Barry E. Williams, May 27 2000
a(22) corrected by T. D. Noe, Feb 08 2008
Maple programs simplified by Johannes W. Meijer, Jun 02 2011

A289780 p-INVERT of the positive integers (A000027), where p(S) = 1 - S - S^2.

Original entry on oeis.org

1, 4, 14, 47, 156, 517, 1714, 5684, 18851, 62520, 207349, 687676, 2280686, 7563923, 25085844, 83197513, 275925586, 915110636, 3034975799, 10065534960, 33382471801, 110713382644, 367182309614, 1217764693607, 4038731742156, 13394504020957, 44423039068114

Offset: 0

Clark Kimberling, Aug 10 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x).
Taking p(S) = 1 - S gives the INVERT transform of s, so that p-INVERT is a generalization of the INVERT transform (e.g., A033453).
Guide to p-INVERT sequences using p(S) = 1 - S - S^2:
t(A000012) = t(1,1,1,1,1,1,1,...)    = A001906
t(A000290) = t(1,4,9,16,25,36,...)   = A289779
t(A000027) = t(1,2,3,4,5,6,7,8,...)  = A289780
t(A000045) = t(1,2,3,5,8,13,21,...)  = A289781
t(A000032) = t(2,1,3,4,7,11,14,...)  = A289782
t(A000244) = t(1,3,9,27,81,243,...)  = A289783
t(A000302) = t(1,4,16,64,256,...)    = A289784
t(A000351) = t(1,5,25,125,625,...)   = A289785
t(A005408) = t(1,3,5,7,9,11,13,...)  = A289786
t(A005843) = t(2,4,6,8,10,12,14,...) = A289787
t(A016777) = t(1,4,7,10,13,16,...)   = A289789
t(A016789) = t(2,5,8,11,14,17,...)   = A289790
t(A008585) = t(3,6,9,12,15,18,...)   = A289795
t(A000217) = t(1,3,6,10,15,21,...)   = A289797
t(A000225) = t(1,3,7,15,31,63,...)   = A289798
t(A000578) = t(1,8,27,64,625,...)    = A289799
t(A000984) = t(1,2,6,20,70,252,...)  = A289800
t(A000292) = t(1,4,10,20,35,56,...)  = A289801
t(A002620) = t(1,2,4,6,9,12,16,...)  = A289802
t(A001906) = t(1,3,8,21,55,144,...)  = A289803
t(A001519) = t(1,1,2,5,13,34,...)    = A289804
t(A103889) = t(2,1,4,3,6,5,8,7,,...) = A289805
t(A008619) = t(1,1,2,2,3,3,4,4,...)  = A289806
t(A080513) = t(1,2,2,3,3,4,4,5,...)  = A289807
t(A133622) = t(1,2,1,3,1,4,1,5,...)  = A289809
t(A000108) = t(1,1,2,5,14,42,...)    = A081696
t(A081696) = t(1,1,3,9,29,97,...)    = A289810
t(A027656) = t(1,0,2,0,3,0,4,0,5...) = A289843
t(A175676) = t(1,0,0,2,0,0,3,0,...)  = A289844
t(A079977) = t(1,0,1,0,2,0,3,...)    = A289845
t(A059841) = t(1,0,1,0,1,0,1,...)    = A289846
t(A000040) = t(2,3,5,7,11,13,...)    = A289847
t(A008578) = t(1,2,3,5,7,11,13,...)  = A289828
t(A000142) = t(1!, 2!, 3!, 4!, ...)  = A289924
t(A000201) = t(1,3,4,6,8,9,11,...)   = A289925
t(A001950) = t(2,5,7,10,13,15,...)   = A289926
t(A014217) = t(1,2,4,6,11,17,29,...) = A289927
t(A000045*) = t(0,1,1,2,3,5,...)     = A289975 (* indicates prepended 0's)
t(A000045*) = t(0,0,1,1,2,3,5,...)   = A289976
t(A000045*) = t(0,0,0,1,1,2,3,5,...) = A289977
t(A290990*) = t(0,1,2,3,4,5,...)     = A290990
t(A290990*) = t(0,0,1,2,3,4,5,...)   = A290991
t(A290990*) = t(0,0,01,2,3,4,5,...)  = A290992

			Example 1:  s = (1,2,3,4,5,6,...) = A000027 and p(S) = 1 - S.
S(x) = x + 2x^2 + 3x^3 + 4x^4 + ...
p(S(x)) = 1 - (x + 2x^2 + 3x^3 + 4x^4 + ... )
- p(0) + 1/p(S(x)) = -1 + 1 + x + 3x^2 + 8x^3 + 21x^4 + ...
T(x) = 1 + 3x + 8x^2 + 21x^3 + ...
t(s) = (1,3,8,21,...) = A001906.
***
Example 2:  s = (1,2,3,4,5,6,...) = A000027 and p(S) = 1 - S - S^2.
S(x) =  x + 2x^2 + 3x^3 + 4x^4 + ...
p(S(x)) = 1 - ( x + 2x^2 + 3x^3 + 4x^4 + ...) - ( x + 2x^2 + 3x^3 + 4x^4 + ...)^2
- p(0) + 1/p(S(x)) = -1 + 1 + x + 4x^2 + 14x^3 + 47x^4 + ...
T(x) = 1 + 4x + 14x^2 + 47x^3 + ...
t(s) = (1,4,14,47,...) = A289780.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -7, 5, -1)

Cf. A000027.

P:=[1,4,14,47];; for n in [5..10^2] do P[n]:=5*P[n-1]-7*P[n-2]+5*P[n-3]-P[n-4]; od; P; # Muniru A Asiru, Sep 03 2017

z = 60; s = x/(1 - x)^2; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289780 *)

x='x+O('x^99); Vec((1-x+x^2)/(1-5*x+7*x^2-5*x^3+x^4)) \\ Altug Alkan, Aug 13 2017

G.f.: (1 - x + x^2)/(1 - 5 x + 7 x^2 - 5 x^3 + x^4).

a(n) = 5*a(n-1) - 7*a(n-2) + 5*a(n-3) - a(n-4).

A133494 Diagonal of the array of iterated differences of A047848.

Original entry on oeis.org

1, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443, 2541865828329, 7625597484987, 22876792454961, 68630377364883

Offset: 0

Paul Barry, Paul Curtz, Dec 23 2007

a(n) is the number of ways to choose a composition C, and then choose a composition of each part of C. - Geoffrey Critzer, Mar 19 2012
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
a(n) is the reptend length of 1/3^(n+1) in decimal. - Jianing Song, Nov 14 2018
Also the number of pairs of integer compositions, the first summing to n and the second with sum equal to the length of the first. If an integer composition is regarded as an arrow from sum to length, these are composable pairs, and the obvious composition operation founds a category of integer compositions. For example, we have (2,1,1,4) . (1,2,1) . (1,2) = (2,6), where dots represent the composition operation. The version without empty compositions is A000244. Composable triples are counted by 1 followed by A000302. The unordered version is A022811. - Gus Wiseman, Jul 14 2022

			From _Gus Wiseman_, Jul 15 2020: (Start)
The a(0) = 1 through a(3) = 9 ways to choose a composition of each part of a composition:
  ()  (1)  (2)      (3)
           (1,1)    (1,2)
           (1),(1)  (2,1)
                    (1,1,1)
                    (1),(2)
                    (2),(1)
                    (1),(1,1)
                    (1,1),(1)
                    (1),(1),(1)
(End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Bishal Deb, Cyclic sieving phenomena via combinatorics of continued fractions, arXiv:2508.13709 [math.CO], 2025. See p. 42.
Index entries for linear recurrences with constant coefficients, signature (3).

The strict version is A336139.
Splittings of partitions are A323583.
Multiset partitions of partitions are A001970.
Partitions of each part of a partition are A063834.
Compositions of each part of a partition are A075900.
Strict partitions of each part of a strict partition are A279785.
Compositions of each part of a strict partition are A304961.
Strict compositions of each part of a composition are A307068.
Compositions of each part of a strict composition are A336127.
Cf. A000012, A000244, A000302, A001906, A006951, A011782, A022811, A071919.
Cf. A072706, A078008, A112467, A182105, A316245, A336128, A336135.
Cf. A339006, A355382, A355384, A355387.

[n eq 0 select 1 else 3^(n-1): n in [0..30]]; // G. C. Greubel, Nov 20 2023

a:= n-> ceil(3^(n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2020

CoefficientList[Series[(1 - 2 x)/(1 - 3 x), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *)
Join[{1}, 3^(Range[0, 30])] (* G. C. Greubel, Nov 20 2023 *)

a(n)=max(1,3^(n-1)) \\ Charles R Greathouse IV, Jul 07 2011

Vec((1-2*x)/(1-3*x) + O(x^100)) \\  Altug Alkan, Oct 30 2015

[(3^n + 2*int(n==0))//3 for n in range(31)] # G. C. Greubel, Nov 20 2023

Binomial transform of A078008. - Paul Curtz, Aug 04 2008
From R. J. Mathar, Nov 11 2008: (Start)
G.f.: (1 - 2*x)/(1 - 3*x).
a(n) = A000244(n-1), n > 0. (End)
From Philippe Deléham, Nov 13 2008: (Start)
a(n) = Sum_{k=0..n} A112467(n,k)*2^k.
a(n) = Sum_{k=0..n} A071919(n,k)*2^k. (End)
Let A(x) be the g.f. Then B(x) = x*A(x) satisfies B(x/(1-x)) = x/(1 - 2*B(x)). - Vladimir Kruchinin, Dec 05 2011
G.f.: 1/(1 - (Sum_{k>=1} (x/(1 - x))^k)). - Joerg Arndt, Sep 30 2012
For n > 0, a(n) = 2*(Sum_{k=0..n-1} a(k)) - 1 = 3^(n-1). - J. Conrad, Oct 29 2015
G.f.: 1 + x/(1 + x)*(1 + 4*x/(1 + 4*x)*(1 + 7*x/(1 + 7*x)*(1 + 10*x/(1 + 10*x)*(1 + .... - Peter Bala, May 27 2017
Invert transform of A011782(n) = 2^(n-1). Second invert transform of A000012. - Gus Wiseman, Jul 19 2020
a(n) = ceiling(3^(n-1)). - Alois P. Heinz, Jul 26 2020
From Elmo R. Oliveira, Mar 31 2025: (Start)
E.g.f.: (2 + exp(3*x))/3.
a(n) = 3*a(n-1) for n > 1. (End)

Definition clarified by R. J. Mathar, Nov 11 2008

A329697 a(n) is the number of iterations needed to reach a power of 2 starting at n and using the map k -> k-(k/p), where p is the largest prime factor of k.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 0, 2, 1, 2, 1, 2, 2, 2, 0, 1, 2, 3, 1, 3, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 0, 3, 1, 3, 2, 3, 3, 3, 1, 2, 3, 4, 2, 3, 3, 4, 1, 4, 2, 2, 2, 3, 3, 3, 2, 4, 3, 4, 2, 3, 3, 4, 0, 3, 3, 4, 1, 4, 3, 4, 2, 3, 3, 3, 3, 4, 3, 4, 1, 4, 2, 3, 3, 2, 4, 4, 2, 3, 3, 4, 3, 4, 4, 4, 1, 2, 4, 4, 2

Offset: 1

Ali Sada and Robert G. Wilson v, Feb 28 2020

From Antti Karttunen, Apr 07 2020: (Start)
Also the least number of iterations of nondeterministic map k -> k-(k/p) needed to reach a power of 2, when any prime factor p of k can be used. The minimal length path to the nearest power of 2 (= 2^A064415(n)) is realized whenever one uses any of the A005087(k) distinct odd prime factors of the current k, at any step of the process. For example, this could be done by iterating with the map k -> k-(k/A078701(k)), i.e., by using the least odd prime factor of k (instead of the largest prime).
Proof: Viewing the prime factorization of changing k as a multiset ("bag") of primes, we see that liquefying any odd prime p with step p -> (p-1) brings at least one more 2 to the bag, while applying p -> (p-1) to any 2 just removes it from the bag, but gives nothing back. Thus the largest (and thus also the nearest) power of 2 is reached by eliminating - step by step - all odd primes from the bag, but none of 2's, and it doesn't matter in which order this is done.
The above implies also that the sequence is totally additive, which also follows because both A064097 and A064415 are. That A064097(n) = A329697(n) + A054725(n) for all n > 1 can be also seen by comparing the initial conditions and the recursion formulas of these three sequences.
For any n, A333787(n) is either the nearest power of 2 reached (= 2^A064415(n)), or occurs on some of the paths from n to there.
(End)
A003401 gives the numbers k where a(k) = A005087(k). See also A336477. - Antti Karttunen, Mar 16 2021

			The trajectory of 15 is {12, 8}, taking 2 iterations to reach 8 = 2^3. So a(15) is 2.
From _Antti Karttunen_, Apr 07 2020: (Start)
Considering all possible paths from 15 to 1 nondeterministic map k -> k-(k/p), where p can be any prime factor of k, we obtain the following graph:
        15
       / \
      /   \
    10     12
    / \   / \
   /   \ /   \
  5     8     6
   \__  |  __/|
      \_|_/   |
        4     3
         \   /
          \ /
           2
           |
           1.
It can be seen that there's also alternative route to 8 via 10 (with 10 = 15-(15/3), where 3 is not the largest prime factor of 15), but it's not any shorter than the route via 12.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..65537
Michael De Vlieger, Annotated fan style binary tree labeling the index n, with a color code where black represents a(n) = 0, red a(n) = 1, and magenta the largest value in a(n) for n = 1..16383.

Cf. A000265, A003401, A005087, A052126, A053575, A054725, A064097, A064415, A078701, A087436, A147545, A171462, A209229, A333123, A333787, A333790, A334107, A334109, A335875, A334204, A335880, A335881, A336396, A336466, A336469 [= a(phi(n))], A336928 [= a(sigma(n))], A336470, A336477, A339970.
Cf. A000079, A334101, A334102, A334103, A334104, A334105, A334106 for positions of 0 .. 6 in this sequence, and also array A334100.
Cf. A334099 (a right inverse, positions of the first occurrence of each n).
Cf. A334091 (first differences), A335429 (partial sums).
Cf. also A331410 (analogous sequence when using the map k -> k + k/p), A334861, A335877 (their sums and differences), see also A335878 and A335884, A335885.

a[n_] := Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, n, # != 2^IntegerExponent[#, 2] &] -1; Array[a, 100]

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1])))); \\ Antti Karttunen, Apr 07 2020

up_to = 2^24;
A329697list(up_to) = { my(v=vector(up_to)); v[1] = 0; for(n=2, up_to, v[n] = if(!bitand(n,n-1),0,1+vecmin(apply(p -> v[n-n/p], factor(n)[, 1]~)))); (v); };
v329697 = A329697list(up_to);
A329697(n) = v329697[n]; \\ Antti Karttunen, Apr 07 2020

A329697(n) = if(n<=2,0, if(isprime(n), A329697(n-1)+1, my(f=factor(n)); (apply(A329697, f[, 1])~ * f[, 2]))); \\ Antti Karttunen, Apr 19 2020

From Antti Karttunen, Apr 07-19 2020: (Start)
a(1) = a(2) = 0; and for n > 2, a(p) = 1 + a(p-1) if p is an odd prime and a(n*m) = a(n) + a(m) if m,n > 1. [This is otherwise equal to the definition of A064097, except here we have a different initial condition, with a(2) = 0].
a(2n) = a(A000265(n)) = a(n).
a(p) = 1+a(p-1), for all odd primes p.
If A209229(n) == 1 [when n is a power of 2], a(n) = 0,
  otherwise a(n) = 1 + a(n-A052126(n)) = 1 + a(A171462(n)).
Equivalently, for non-powers of 2, a(n) = 1 + a(n-(n/A078701(n))),
or equivalently, for non-powers of 2, a(n) = 1 + Min a(n - n/p), for p prime and dividing n.
a(n) = A064097(n) - A064415(n), or equally, a(n) = A064097(n) - A054725(n), for n > 1.
a(A019434(n)) = 1, a(A334092(n)) = 2, a(A334093(n)) = 3, etc. for all applicable n.
For all n >= 0, a(A334099(n)) = a(A000244(n)) = a(A000351(n)) = a(A001026(n)) = a(257^n) = a(65537^n) = n.
a(A122111(n)) = A334107(n), a(A225546(n)) = A334109(n).
(End)
From Antti Karttunen, Mar 16 2021: (Start)
a(n) = a(A336466(n)) + A087436(n) = A336396(n) + A087436(n).
a(A053575(n)) = A336469(n) = a(n) - A005087(n).
a(A147545(n)) = A000120(A147545(n)) - 1.
(End)

A027471 a(n) = (n-1)*3^(n-2), n > 0.

Original entry on oeis.org

0, 1, 6, 27, 108, 405, 1458, 5103, 17496, 59049, 196830, 649539, 2125764, 6908733, 22320522, 71744535, 229582512, 731794257, 2324522934, 7360989291, 23245229340, 73222472421, 230127770466, 721764371007, 2259436291848

Offset: 1

Olivier Gérard

Arithmetic derivative of 3^(n-1): a(n) = A003415(A000244(n-1)). - Reinhard Zumkeller, Feb 26 2002 [Offset corrected by Jianing Song, May 28 2024]
Binomial transform of A053220(n+1) is a(n+2). Binomial transform of A001787 is a(n+1). Binomial transform of A045883(n-1). - Michael Somos, Jul 10 2003
If X_1,X_2,...,X_n are 3-blocks of a (3n+1)-set X then, for n >= 1, a(n+2) is the number of (n+1)-subsets of X intersecting each X_i, (i=1,2,...,n). > - Milan Janjic, Nov 18 2007
Let S be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xSy if x is a subset of y. Then a(n+1) = the sum of the differences in size (i.e., |y|-|x|) for all (x, y) of S. - Ross La Haye, Nov 19 2007
Number of substrings 00 (or 11, or 22) in all ternary words of length n: a(3) = 6 because we have 000, 001, 002, 100, 200 (with 000 contributing two substrings). - Darrell Minor, Jul 17 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..700
Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Descent distribution on Catalan words avoiding a pattern of length at most three, arXiv:1803.06706 [math.CO], 2018.
Samuele Giraudo, Pluriassociative algebras I: The pluriassociative operad, arXiv:1603.01040 [math.CO], 2016.
Milan Janjić, Two Enumerative Functions
Milan Janjić and Boris Petković, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Milan Janjić and Boris Petković, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Frank Ellermann, Illustration of binomial transforms
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 715
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.
Aleksandar Petojević, A Note about the Pochhammer Symbol, Mathematica Moravica, Vol. 12-1 (2008), 37-42.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Mark Shattuck, Enumeration of consecutive patterns in flattened Catalan words, arXiv:2502.10661 [math.CO], 2025. See pp. 3, 20.
Index entries for linear recurrences with constant coefficients, signature (6,-9).

Second column of A027465.
Partial sums of A081038.
Cf. A006234.

List([1..40], n-> (n-1)*3^(n-2)); # Muniru A Asiru, Jul 15 2018

[(n-1)*3^(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 09 2011

seq((n-1)*3^(n-2), n=1..40); # Muniru A Asiru, Jul 15 2018

Table[(n-1)3^(n-2),{n,30}] (* or *)
LinearRecurrence[{6,-9},{0,1},30] (* Harvey P. Dale, Apr 14 2016 *)
Range[0, 24]! CoefficientList[ Series[x*Exp[3 x], {x, 0, 24}], x] (* Robert G. Wilson v, Aug 03 2018 *)

PARI
```
a(n)=if(n<1, 0, (n-1)*3^(n-2));
```

[3^(n-2)*(n-1) for n in (1..30)] # G. C. Greubel, May 20 2021

From Wolfdieter Lang: (Start)
G.f.: (x/(1-3*x))^2.
E.g.f.: (1 + (3*x-1)*exp(3*x))/9.
a(n) = 3^(n-2)*(n-1) (convolution of A000244, powers of 3, with itself). (End)
a(n) = 6*a(n-1) - 9*a(n-2), n > 2, a(1)=0, a(2)=1. - Barry E. Williams, Jan 13 2000
a(n) = A036290(n-1)/3, for n>0. - Paul Barry, Feb 06 2004 [corrected by Jerzy R Borysowicz, Apr 03 2025]
a(n) = Sum_{k=0..n} 3^(n-k)*binomial(n-k+1, k)*binomial(1, (k+1)/2)*(1-(-1)^k)/2.
From Paul Barry, Feb 15 2005: (Start)
a(n) = (1/3)*Sum_{k=0..2n} T(n, k)*k, where T(n, k) is given by A027907.
a(n) = (1/3)*Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j+k).
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j-k).
a(n+1) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*(j+k+1). (End)
Sum_{n>=2} 1/a(n) = 3*log(3/2). - Jaume Oliver Lafont, Sep 19 2009
a(n) = 3*a(n-1) + 3^(n-2) (with a(1)=0). - Vincenzo Librandi, Dec 30 2010
Sum_{n>=2} (-1)^n/a(n) = 3*log(4/3). - Amiram Eldar, Oct 28 2020

Edited by Michael Somos, Jul 10 2003

A001021 Powers of 12.

Original entry on oeis.org

1, 12, 144, 1728, 20736, 248832, 2985984, 35831808, 429981696, 5159780352, 61917364224, 743008370688, 8916100448256, 106993205379072, 1283918464548864, 15407021574586368, 184884258895036416, 2218611106740436992

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 12), L(1, 12), P(1, 12), T(1, 12). Essentially same as Pisot sequences E(12, 144), L(12, 144), P(12, 144), T(12, 144). See A008776 for definitions of Pisot sequences.
Central terms of the triangle in A100851. - Reinhard Zumkeller, Mar 04 2006
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=1, a(n) equals the number of 12-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Starting with 12, this sequence appears in the film "Vollmond" (1998, dir. Fredi Murer), when the children write it on the sidewalk at night. - Alonso del Arte, Dec 21 2011

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 276.
Tanya Khovanova, Recursive Sequences.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Index entries for linear recurrences with constant coefficients, signature (12).

Cf. A024140 and A178248, 12^n +/- 1; A016125.

Cf. A000005, A000244, A000302, A000351, A000384, A001222, A008585, A008776, A100851, A159991.

a001021 = (12 ^)
a001021_list = iterate (* 12) 1  -- Reinhard Zumkeller, Mar 31 2012

A001021:=-1/(-1+12*z); # Simon Plouffe in his 1992 dissertation

Table[12^n, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

a(n)=12^n \\ Charles R Greathouse IV, Dec 21 2011

G.f.: 1/(1-12*x).
E.g.f.: exp(12x).
a(n) = 12*a(n-1). - Zerinvary Lajos, Apr 27 2009
a(n) = A159991(n)/A000351(n). - Reinhard Zumkeller, May 02 2009
From Reinhard Zumkeller, Mar 31 2012: (Start)
a(n) = A000302(n) * A000244(n). - Reinhard Zumkeller, Mar 31 2012
A001222(a(n)) = A008585(n); A000005(a(n)) = A000384(a(n)). (End)
a(n) = det(|ps(i+2, j)|, 1 <= i, j <= n), where ps(n, k) are Legendre-Stirling numbers of the first kind. - Mircea Merca, Apr 04 2013

A013609 Triangle of coefficients in expansion of (1+2*x)^n.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 1, 6, 12, 8, 1, 8, 24, 32, 16, 1, 10, 40, 80, 80, 32, 1, 12, 60, 160, 240, 192, 64, 1, 14, 84, 280, 560, 672, 448, 128, 1, 16, 112, 448, 1120, 1792, 1792, 1024, 256, 1, 18, 144, 672, 2016, 4032, 5376, 4608, 2304, 512, 1, 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024

Offset: 0

N. J. A. Sloane

T(n,k) is the number of lattice paths from (0,0) to (n,k) with steps (1,0) and two kinds of steps (1,1). The number of paths with steps (1,0) and s kinds of steps (1,1) corresponds to the expansion of (1+s*x)^n. - Joerg Arndt, Jul 01 2011
Also sum of rows in A046816. - Lior Manor, Apr 24 2004
Also square array of unsigned coefficients of Chebyshev polynomials of second kind. - Philippe Deléham, Aug 12 2005
The rows give the number of k-simplices in the n-cube. For example, 1, 6, 12, 8 shows that the 3-cube has 1 volume, 6 faces, 12 edges and 8 vertices. - Joshua Zucker, Jun 05 2006
Triangle whose (i, j)-th entry is binomial(i, j)*2^j.
With offset [1,1] the triangle with doubled numbers, 2*a(n,m), enumerates sequences of length m with nonzero integer entries n_i satisfying sum(|n_i|) <= n. Example n=4, m=2: [1,3], [3,1], [2,2] each in 2^2=4 signed versions: 2*a(4,2) = 2*6 = 12. The Sum over m (row sums of 2*a(n,m)) gives 2*3^(n-1), n >= 1. See the W. Lang comment and a K. A. Meissner reference under A024023. - Wolfdieter Lang, Jan 21 2008
n-th row of the triangle = leftmost column of nonzero terms of X^n, where X = an infinite bidiagonal matrix with (1,1,1,...) in the main diagonal and (2,2,2,...) in the subdiagonal. - Gary W. Adamson, Jul 19 2008
Numerators of a matrix square-root of Pascal's triangle A007318, where the denominators for the n-th row are set to 2^n. - Gerald McGarvey, Aug 20 2009
From Johannes W. Meijer, Sep 22 2010: (Start)
The triangle sums (see A180662 for their definitions) link the Pell-Jacobsthal triangle, whose mirror image is A038207, with twenty-four different sequences; see the crossrefs.
This triangle may very well be called the Pell-Jacobsthal triangle in view of the fact that A000129 (Kn21) are the Pell numbers and A001045 (Kn11) the Jacobsthal numbers.
(End)
T(n,k) equals the number of n-length words on {0,1,2} having n-k zeros. - Milan Janjic, Jul 24 2015
T(n-1,k-1) is the number of 2-compositions of n with zeros having k positive parts; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 16 2020
T(n,k) is the number of chains 0=x_0Geoffrey Critzer, Oct 01 2022
Excluding the initial 1, T(n,k) is the number of k-faces of a regular n-cross polytope. See A038207 for n-cube and A135278 for n-simplex. - Mohammed Yaseen, Jan 14 2023

			Triangle begins:
  1;
  1,  2;
  1,  4,   4;
  1,  6,  12,    8;
  1,  8,  24,   32,   16;
  1, 10,  40,   80,   80,    32;
  1, 12,  60,  160,  240,   192,    64;
  1, 14,  84,  280,  560,   672,   448,    128;
  1, 16, 112,  448, 1120,  1792,  1792,   1024,    256;
  1, 18, 144,  672, 2016,  4032,  5376,   4608,   2304,    512;
  1, 20, 180,  960, 3360,  8064, 13440,  15360,  11520,   5120,  1024;
  1, 22, 220, 1320, 5280, 14784, 29568,  42240,  42240,  28160, 11264,  2048;
  1, 24, 264, 1760, 7920, 25344, 59136, 101376, 126720, 112640, 67584, 24576, 4096;
From _Peter Bala_, Apr 20 2012: (Start)
The triangle can be written as the matrix product A038207*(signed version of A013609).
  |.1................||.1..................|
  |.2...1............||-1...2..............|
  |.4...4...1........||.1..-4...4..........|
  |.8..12...6...1....||-1...6...-12...8....|
  |16..32..24...8...1||.1..-8....24.-32..16|
  |..................||....................|
(End)

B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
G. Hotz, Zur Reduktion von Schaltkreispolynomen im Hinblick auf eine Verwendung in Rechenautomaten, El. Datenverarbeitung, Folge 5 (1960), pp. 21-27.

T. D. Noe, Rows n=0..50 of triangle, flattened
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.
H. J. Brothers, Pascal's Prism: Supplementary Material, PDF version.
John Cartan, Cartan's triangle shows the relationship to the n-cube.
R. Ehrenborg and M. Readdy, Characterization of the factorial functions of Eulerian binomial and Sheffer posets
J. Goldman and J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 1-rook coefficients of k rooks on the 2xn board, all heights 2.
W. G. Harter, Representations of multidimensional symmetries in networks, J. Math. Phys., 15 (1974), 2016-2021.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
D. A. Zaitsev, A generalized neighborhood for cellular automata, Theoretical Computer Science, 666 (2017), 21-35.

Cf. A007318, A013610, etc.
Appears in A167580 and A167591. - Johannes W. Meijer, Nov 23 2009
From Johannes W. Meijer, Sep 22 2010: (Start)
Triangle sums (see the comments): A000244 (Row1); A000012 (Row2); A001045 (Kn11); A026644 (Kn12); 4*A011377 (Kn13); A000129 (Kn21); A094706 (Kn22); A099625 (Kn23); A001653 (Kn3); A007583 (Kn4); A046717 (Fi1); A007051 (Fi2); A077949 (Ca1); A008998 (Ca2); A180675 (Ca3); A092467 (Ca4); A052942 (Gi1); A008999 (Gi2); A180676 (Gi3); A180677 (Gi4); A140413 (Ze1); A180678 (Ze2); A097117 (Ze3); A055588 (Ze4).
(End)
Cf. A024023, A038207, A046816, A105728, A115068, A135278, A171977, A180662.
T(2n,n) gives A059304.

a013609 n = a013609_list !! n
a013609_list = concat $ iterate ([1,2] *) [1]
instance Num a => Num [a] where
   fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

a013609 n k = a013609_tabl !! n !! k
a013609_row n = a013609_tabl !! n
a013609_tabl = iterate (\row -> zipWith (+) ([0] ++ row) $
                                zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Jul 22 2013, Feb 27 2013

[2^k*Binomial(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Sep 17 2021

bin2:=proc(n,k) option remember; if k<0 or k>n then 0 elif k=0 then 1 else 2*bin2(n-1,k-1)+bin2(n-1,k); fi; end; # N. J. A. Sloane, Jun 01 2009

Flatten[Table[CoefficientList[(1 + 2*x)^n, x], {n, 0, 10}]][[1 ;; 59]] (* Jean-François Alcover, May 17 2011 *)
BinomialROW[n_, k_, t_] := Sum[Binomial[n, k]*Binomial[k, j]*(-1)^(k - j)*t^j, {j, 0, k}]; Column[Table[BinomialROW[n, k, 3], {n, 0, 10}, {k, 0, n}], Center] (* Kolosov Petro, Jan 28 2019 *)

a(n,k):=coeff(expand((1+2*x)^n),x^k);
create_list(a(n,k),n,0,6,k,0,n); /* Emanuele Munarini, Nov 21 2012 */

/* same as in A092566 but use */
steps=[[1,0], [1,1], [1,1]]; /* note double [1,1] */
/* Joerg Arndt, Jul 01 2011 */

flatten([[2^k*binomial(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Sep 17 2021

G.f.: 1 / (1 - x*(1+2*y)).
T(n,k) = 2^k*binomial(n,k).
T(n,k) = 2*T(n-1,k-1) + T(n-1,k). - Jon Perry, Nov 22 2005
Row sums are 3^n = A000244(n). - Joerg Arndt, Jul 01 2011
T(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i). - Mircea Merca, Apr 28 2012
E.g.f.: exp(2*y*x + x). - Geoffrey Critzer, Nov 12 2012
Riordan array (x/(1 - x), 2*x/(1 - x)). Exp(2*x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(2*x)*(1 + 6*x + 12*x^2/2! + 8*x^3/3!) = 1 + 8*x + 40*x^2/2! + 160*x^3/3! + 560*x^4/4! + .... The same property holds more generally for Riordan arrays of the form (f(x), 2*x/(1 - x)). - Peter Bala, Dec 21 2014
T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(n,k) * binomial(k,j) * 3^j. - Kolosov Petro, Jan 28 2019
T(n,k) = 2*(n+1-k)*T(n,k-1)/k, T(n,0) = 1. - Alexander R. Povolotsky, Oct 08 2023
For n >= 1, GCD(T(n,1), ..., T(n,n)) = GCD(T(n,1),T(n,n)) = GCD(2*n,2^n) = A171977(n). - Pontus von Brömssen, Nov 01 2024

A083420 a(n) = 2*4^n - 1.

Original entry on oeis.org

1, 7, 31, 127, 511, 2047, 8191, 32767, 131071, 524287, 2097151, 8388607, 33554431, 134217727, 536870911, 2147483647, 8589934591, 34359738367, 137438953471, 549755813887, 2199023255551, 8796093022207, 35184372088831, 140737488355327, 562949953421311

Offset: 0

Paul Barry, Apr 29 2003

Sum of divisors of 4^n. - Paul Barry, Oct 13 2005
Subsequence of A000069; A132680(a(n)) = A005408(n). - Reinhard Zumkeller, Aug 26 2007
If x = a(n), y = A000079(n+1) and z = A087289(n), then x^2 + 2*y^2 = z^2. - Vincenzo Librandi, Jun 09 2014
It seems that a(n) divides A001676(3+4n). Several other entries apparently have this sequence embedded in them, e.g., A014551, A168604, A213243, A213246-8, and A279872. - Tom Copeland, Dec 27 2016
To elaborate on Librandi's comment from 2014: all these numbers, even if prime in Z, are sure not to be prime in Z[sqrt(2)], since a(n) can at least be factored as ((2^(2n + 1) - 1) - (2^(2n) - 1)*sqrt(2))((2^(2n + 1) - 1) + (2^(2n) - 1)*sqrt(2)). For example, 7 = (3 - sqrt(2))(3 + sqrt(2)), 31 = (7 - 3*sqrt(2))(7 + 3*sqrt(2)), 127 = (15 - 7*sqrt(2))(15 + 7*sqrt(2)). - Alonso del Arte, Oct 17 2017
Largest odd factors of A147590. - César Aguilera, Jan 07 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Roudy El Haddad, Recurrent Sums and Partition Identities, arXiv:2101.09089 [math.NT], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 1: Recurrent sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata with even rule numbers, Fig 11.
Robert Schneider, Partition zeta functions, Research in Number Theory, 2(1):9, 2016.
Eric Weisstein's World of Mathematics, Rule 220
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A083421, A000668 (primes in this sequence), A004171, A000244.
Cf. A001676, A014551, A168604, A213243, A213246, A213247, A213248, A279872.
Cf. A000302.

a083420 = subtract 1 . (* 2) . (4 ^)  -- Reinhard Zumkeller, Dec 22 2015

[2*4^n-1 : n in [0..30]]; // Wesley Ivan Hurt, Mar 14 2015

seq(2*4^n-1, n = 0..22); # Peter Luschny, Aug 17 2011

2 * 4^Range[0, 31] - 1 (* Alonso del Arte, Oct 17 2017 *)

a(n)=2*4^n-1 \\ Charles R Greathouse IV, Sep 24 2015

def A083420(n): return (1<<(n<<1|1))-1 # Chai Wah Wu, Mar 10 2025

G.f.: (1+2*x)/((1-x)*(1-4*x)).
E.g.f.: 2*exp(4*x)-exp(x).
With a leading zero, this is a(n) = (4^n - 2 + 0^n)/2, the binomial transform of A080925. - Paul Barry, May 19 2003
From Benoit Cloitre, Jun 18 2004: (Start)
a(n) = (-16^n/2)*B(2n, 1/4)/B(2n) where B(n, x) is the n-th Bernoulli polynomial and B(k) = B(k, 0) is the k-th Bernoulli number.
a(n) = 5*a(n-1) - 4*a(n-2).
a(n) = (-4^n/2)*B(2*n, 1/2)/B(2*n). (End)
a(n) = A099393(n) + A020522(n) = A000302(n) + A024036(n). - Reinhard Zumkeller, Feb 07 2006
a(n) = Stirling2(2*(n+1), 2). - Zerinvary Lajos, Dec 06 2006
a(n) = 4*a(n-1) + 3 with n > 0, a(0) = 1. - Vincenzo Librandi, Dec 30 2010
a(n) = A001576(n+1) - 2*A001576(n). - Brad Clardy, Mar 26 2011
a(n) = 6*A002450(n) + 1. - Roderick MacPhee, Jul 06 2012
a(n) = A000203(A000302(n)). - Michel Marcus, Jan 20 2014
a(n) = Sum_{i = 0..n} binomial(2n+2, 2i). - Wesley Ivan Hurt, Mar 14 2015
a(n) = (1/4^n) * Sum_{k = 0..n} binomial(2*n+1,2*k)*9^k. - Peter Bala, Feb 06 2019
a(n) = A147590(n)/A000079(n). - César Aguilera, Jan 07 2020
a(n) = numerator(zeta_star({2}(n + 1))/zeta(2*n + 2)) where zeta_star is the multiple zeta star values and ({2}_n) represents (2, ..., 2) where the multiplicity of 2 is n. - _Roudy El Haddad, Feb 22 2022

cp's OEIS Frontend

A002379 a(n) = floor(3^n / 2^n).

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

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A025192 a(0)=1; a(n) = 2*3^(n-1) for n >= 1.

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A289780 p-INVERT of the positive integers (A000027), where p(S) = 1 - S - S^2.

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A133494 Diagonal of the array of iterated differences of A047848.

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