A003953 -id:A003953 - cp's OEIS frontend

A007283 a(n) = 3*2^n.

3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944, 12884901888

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(3, 6), L(3, 6), P(3, 6), T(3, 6). See A008776 for definitions of Pisot sequences.
Numbers k such that A006530(A000010(k)) = A000010(A006530(k)) = 2. - Labos Elemer, May 07 2002
Also least number m such that 2^n is the smallest proper divisor of m which is also a suffix of m in binary representation, see A080940. - Reinhard Zumkeller, Feb 25 2003
Length of the period of the sequence Fibonacci(k) (mod 2^(n+1)). - Benoit Cloitre, Mar 12 2003
The sequence of first differences is this sequence itself. - Alexandre Wajnberg and Eric Angelini, Sep 07 2005
Subsequence of A122132. - Reinhard Zumkeller, Aug 21 2006
Apart from the first term, a subsequence of A124509. - Reinhard Zumkeller, Nov 04 2006
Total number of Latin n-dimensional hypercubes (Latin polyhedra) of order 3. - Kenji Ohkuma (k-ookuma(AT)ipa.go.jp), Jan 10 2007
Number of different ternary hypercubes of dimension n. - Edwin Soedarmadji (edwin(AT)systems.caltech.edu), Dec 10 2005
For n >= 1, a(n) is equal to the number of functions f:{1, 2, ..., n + 1} -> {1, 2, 3} such that for fixed, different x_1, x_2,...,x_n in {1, 2, ..., n + 1} and fixed y_1, y_2,...,y_n in {1, 2, 3} we have f(x_i) <> y_i, (i = 1,2,...,n). - Milan Janjic, May 10 2007
a(n) written in base 2: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, n times 0 (see A003953). - Jaroslav Krizek, Aug 17 2009
Subsequence of A051916. - Reinhard Zumkeller, Mar 20 2010
Numbers containing the number 3 in their Collatz trajectories. - Reinhard Zumkeller, Feb 20 2012
a(n-1) gives the number of ternary numbers with n digits with no two adjacent digits in common; e.g., for n=3 we have 010, 012, 020, 021, 101, 102, 120, 121, 201, 202, 210 and 212. - Jon Perry, Oct 10 2012
If n > 1, then a(n) is a solution for the equation sigma(x) + phi(x) = 3x-4. This equation also has solutions 84, 3348, 1450092, ...  which are not of the form 3*2^n. - Farideh Firoozbakht, Nov 30 2013
a(n) is the upper bound for the "X-ray number" of any convex body in E^(n + 2), conjectured by Bezdek and Zamfirescu, and proved in the plane E^2 (see the paper by Bezdek and Zamfirescu). - L. Edson Jeffery, Jan 11 2014
If T is a topology on a set V of size n and T is not the discrete topology, then T has at most 3 * 2^(n-2) many open sets. See Brown and Stephen references. - Ross La Haye, Jan 19 2014
Comment from Charles Fefferman, courtesy of Doron Zeilberger, Dec 02 2014: (Start)
Fix a dimension n. For a real-valued function f defined on a finite set E in R^n, let Norm(f, E) denote the inf of the C^2 norms of all functions F on R^n that agree with f on E. Then there exist constants k and C depending only on the dimension n such that Norm(f, E) <= C*max{ Norm(f, S) }, where the max is taken over all k-point subsets S in E.  Moreover, the best possible k is 3 * 2^(n-1).
The analogous result, with the same k, holds when the C^2 norm is replaced, e.g., by the C^1, alpha norm (0 < alpha <= 1). However, the optimal analogous k, e.g., for the C^3 norm is unknown.
For the above results, see Y. Brudnyi and P. Shvartsman (1994). (End)
Also, coordination sequence for (infinity, infinity, infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015
The average of consecutive powers of 2 beginning with 2^1. - Melvin Peralta and Miriam Ong Ante, May 14 2016
For n > 1, a(n) is the smallest Zumkeller number with n divisors that are also Zumkeller numbers (A083207). - Ivan N. Ianakiev, Dec 09 2016
Also, for n >= 2, the number of length-n strings over the alphabet {0,1,2,3} having only the single letters as nonempty palindromic subwords. (Corollary 21 in Fleischer and Shallit) - Jeffrey Shallit, Dec 02 2019
Also, a(n) is the minimum link-length of any covering trail, circuit, path, and cycle for the set of the 2^(n+2) vertices of an (n+2)-dimensional hypercube. - Marco Ripà, Aug 22 2022
The known fixed points of maps n -> A163511(n) and n -> A243071(n). [See comments in A163511]. - Antti Karttunen, Sep 06 2023
The finite subsequence a(3), a(4), a(5), a(6) = 24, 48, 96, 192 is one of only two geometric sequences that can be formed with all interior angles (all integer, in degrees) of a simple polygon. The other sequence is a subsequence of A000244 (see comment there). - Felix Huber, Feb 15 2024
A level 1 Sierpiński triangle is a triangle. Level n+1 is formed from three copies of level n by identifying pairs of corner vertices of each pair of triangles.  For n>2, a(n-3) is the radius of the level n Sierpiński triangle graph. - Allan Bickle, Aug 03 2024

Jason I. Brown, Discrete Structures and Their Interactions, CRC Press, 2013, p. 71.
T. Ito, Method, equipment, program and storage media for producing tables, Publication number JP2004-272104A, Japan Patent Office (written in Japanese, a(2)=12, a(3)=24, a(4)=48, a(5)=96, a(6)=192, a(7)=384 (a(7)=284 was corrected)).
Kenji Ohkuma, Atsuhiro Yamagishi and Toru Ito, Cryptography Research Group Technical report, IT Security Center, Information-Technology Promotion Agency, JAPAN.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
K. Bezdek and Tudor Zamfirescu, A Characterization of 3-dimensional Convex Sets with an Infinite X-ray Number, in: Coll. Math. Soc. J. Bolyai 63., Intuitive Geometry, Szeged (Hungary), North-Holland, Amsterdam, 1991, pp. 33-38.
Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
Yuri Brudnyi and Pavel Shvartsman, Generalizations of Whitney's extension theorem, International Mathematics Research Notices 1994.3 (1994): 129-139.
J. W. Cannon and P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Tomislav Došlić, Kepler-Bouwkamp Radius of Combinatorial Sequences, Journal of Integer Sequences, Vol. 17, 2014, #14.11.3.
John Elias, Illustration: 2^n+1 hexagram perimeters
Lukas Fleischer and Jeffrey Shallit, Words With Few Palindromes, Revisited, arxiv preprint arXiv:1911.12464 [cs.FL], November 27 2019.
A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
Tanya Khovanova, Recursive Sequences
Roberto Rinaldi and Marco Ripà, Optimal cycles enclosing all the nodes of a k-dimensional hypercube, arXiv:2212.11216 [math.CO], 2022.
Edwin Soedarmadji, Latin Hypercubes and MDS Codes, Discrete Mathematics, Volume 306, Issue 12, Jun 28 2006, Pages 1232-1239
D. Stephen, Topology on Finite Sets, American Mathematical Monthly, 75: 739 - 741, 1968.
Index entries for linear recurrences with constant coefficients, signature (2).

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.
Subsequence of the following sequences: A029744, A029747, A029748, A029750, A362804 (after 3), A364494, A364496, A364289, A364291, A364292, A364295, A364497, A364964, A365422.
Essentially same as A003945 and A042950.
Row sums of (5, 1)-Pascal triangle A093562 and of (1, 5) Pascal triangle A096940.
Cf. A000079, A100540, A124508, A221718.
Cf. Latin squares: A000315, A002860, A003090, A040082, A003191; Latin cubes: A098843, A098846, A098679, A099321.
Cf. A133466, A225546, A163511, A243071.
Cf. A007283, A029858, A067771, A193277, A233774, A233775, A246959.

a007283 = (* 3) . (2 ^)
a007283_list = iterate (* 2) 3
-- Reinhard Zumkeller, Mar 18 2012, Feb 20 2012

[3*2^n: n in [0..30]]; // Vincenzo Librandi, May 18 2011

A007283:=n->3*2^n; seq(A007283(n), n=0..50); # Wesley Ivan Hurt, Dec 03 2013

Table[3(2^n), {n, 0, 32}] (* Alonso del Arte, Mar 24 2011 *)

A007283(n):=3*2^n$
makelist(A007283(n),n,0,30); /* Martin Ettl, Nov 11 2012 */

PARI
```
a(n)=3*2^n
```

a(n)=3<Charles R Greathouse IV, Oct 10 2012

def A007283(n): return 3<Chai Wah Wu, Feb 14 2023

(List.fill(40)(2: BigInt)).scanLeft(1: BigInt)( * ).map(3 * ) // _Alonso del Arte, Nov 28 2019

G.f.: 3/(1-2*x).
a(n) = 2*a(n - 1), n > 0; a(0) = 3.
a(n) = Sum_{k = 0..n} (-1)^(k reduced (mod 3))*binomial(n, k). - Benoit Cloitre, Aug 20 2002
a(n) = A118416(n + 1, 2) for n > 1. - Reinhard Zumkeller, Apr 27 2006
a(n) = A000079(n) + A000079(n + 1). - Zerinvary Lajos, May 12 2007
a(n) = A000079(n)*3. - Omar E. Pol, Dec 16 2008
From Paul Curtz, Feb 05 2009: (Start)
a(n) = b(n) + b(n+3) for b = A001045, A078008, A154879.
a(n) = abs(b(n) - b(n+3)) with b(n) = (-1)^n*A084247(n). (End)
a(n) = 2^n + 2^(n + 1). - Jaroslav Krizek, Aug 17 2009
a(n) = A173786(n + 1, n) = A173787(n + 2, n). - Reinhard Zumkeller, Feb 28 2010
A216022(a(n)) = 6 and A216059(a(n)) = 7, for n > 0. - Reinhard Zumkeller, Sep 01 2012
a(n) = (A000225(n) + 1)*3. - Martin Ettl, Nov 11 2012
E.g.f.: 3*exp(2*x). - Ilya Gutkovskiy, May 15 2016
A020651(a(n)) = 2. - Yosu Yurramendi, Jun 01 2016
a(n) = sqrt(A014551(n + 1)*A014551(n + 2) + A014551(n)^2). - Ezhilarasu Velayutham, Sep 01 2019
a(A048672(n)) = A225546(A133466(n)). - Michel Marcus and Peter Munn, Nov 29 2019
Sum_{n>=1} 1/a(n) = 2/3. - Amiram Eldar, Oct 28 2020

A003945 Expansion of g.f. (1+x)/(1-2*x).

Original entry on oeis.org

1, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944, 12884901888

Offset: 0

N. J. A. Sloane

Coordination sequence for infinite tree with valency 3.
Number of Hamiltonian cycles in K_3 X P_n.
Number of ternary words of length n avoiding aa, bb, cc.
For n > 0, row sums of A029635. - Paul Barry, Jan 30 2005
Binomial transform is {1, 4, 13, 40, 121, 364, ...}, see A003462. - Philippe Deléham, Jul 23 2005
Convolved with the Jacobsthal sequence A001045 = A001786: (1, 4, 12, 32, 80, ...). - Gary W. Adamson, May 23 2009
Equals (n+1)-th row sums of triangle A161175. - Gary W. Adamson, Jun 05 2009
a(n) written in base 2: a(0) = 1, a(n) for n >= 1: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, (n-1) times 0 (see A003953(n)). - Jaroslav Krizek, Aug 17 2009
INVERTi transform of A003688. - Gary W. Adamson, Aug 05 2010
An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 42, 138, 162 and 168, lead to this sequence. For the corner squares these vectors lead to the companion sequence A083329. - Johannes W. Meijer, Aug 15 2010
A216022(a(n)) != 2 and A216059(a(n)) != 3. - Reinhard Zumkeller, Sep 01 2012
Number of length-n strings of 3 letters with no two adjacent letters identical. The general case (strings of r letters) is the sequence with g.f. (1+x)/(1-(r-1)*x). - Joerg Arndt, Oct 11 2012
Sums of pairs of rows of Pascal's triangle A007318, T(2n,k)+T(2n+1,k); Sum_{n>=1} A000290(n)/a(n) = 4. - John Molokach, Sep 26 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Yasemin Alp and E. Gokcen Kocer, Exponential Almost-Riordan Arrays, Results Math. (2024) Vol. 79, 173.
F. Faase, Counting Hamiltonian cycles in product graphs
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 151
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 304
Markus Kuba and Alois Panholzer, Enumeration formulas for pattern restricted Stirling permutations, Discrete Math. 312 (2012), no. 21, 3179--3194. MR2957938. - From _N. J. A. Sloane_, Sep 25 2012
C. Richard and U. Grimm, On the entropy and letter frequencies of ternary squarefree words, arXiv:math/0302302 [math.CO], 2003.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (2).
Index entries for sequences related to trees

Essentially same as A007283 (3*2^n) and A042950.
Cf. A001787, A001045, A161175.
Generating functions of the form (1+x)/(1-k*x) for k=1 to 12: A040000, A003945, A003946, A003947, A003948, A003949, A003950, A003951, A003952.
Generating functions of the form (1+x)/(1-k*x) for k=13 to 30: A170732, A170733, A170734, A170735, A170736, A170737, A170738, A170739, A170740, A170741, A170742, A170743, A170744, A170745, A170746, A170747, A170748.
Generating functions of the form (1+x)/(1-k*x) for k=31 to 50: A170749, A170750, A170751, A170752, A170753, A170754, A170755, A170756, A170757, A170758, A170759, A170760, A170761, A170762, A170763, A170764, A170765, A170766, A170767, A170768, A170769.
Cf. A003688.

k := 3; if n = 0 then 1 else k*(k-1)^(n-1); fi;

Join[{1}, 3*2^Range[0, 60]] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
Table[2^n+Floor[2^(n-1)], {n,0,30}] (* Martin Grymel, Oct 17 2012 *)
CoefficientList[Series[(1+x)/(1-2x),{x,0,40}],x] (* or *) LinearRecurrence[ {2},{1,3},40] (* Harvey P. Dale, May 04 2017 *)

a(n)=if(n,3<Charles R Greathouse IV, Jan 12 2012

a(0) = 1; for n > 0, a(n) = 3*2^(n-1).
a(n) = 2*a(n-1), n > 1; a(0)=1, a(1)=3.
More generally, the g.f. (1+x)/(1-k*x) produces the sequence [1, 1 + k, (1 + k)*k, (1 + k)*k^2, ..., (1+k)*k^(n-1), ...], with a(0) = 1, a(n) = (1+k)*k^(n-1) for n >= 1. Also a(n+1) = k*a(n) for n >= 1. - Zak Seidov and N. J. A. Sloane, Dec 05 2009
The g.f. (1+x)/(1-k*x) produces the sequence with closed form (in PARI notation) a(n)=(n>=0)*k^n+(n>=1)*k^(n-1). - Jaume Oliver Lafont, Dec 05 2009
Binomial transform of A000034. a(n) = (3*2^n - 0^n)/2. - Paul Barry, Apr 29 2003
a(n) = Sum_{k=0..n} (n+k)*binomial(n, k)/n. - Paul Barry, Jan 30 2005
a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 1. - Philippe Deléham, Jul 10 2005
Binomial transform of A000034. Hankel transform is {1,-3,0,0,0,...}. - Paul Barry, Aug 29 2006
a(0) = 1, a(n) = 2 + Sum_{k=0..n-1} a(k) for n >= 1. - Joerg Arndt, Aug 15 2012
a(n) = 2^n + floor(2^(n-1)). - Martin Grymel, Oct 17 2012
E.g.f.: (3*exp(2*x) - 1)/2. - Stefano Spezia, Jan 31 2023

Edited by N. J. A. Sloane, Dec 04 2009

A003954 Expansion of g.f.: (1+x)/(1-11*x).

Original entry on oeis.org

1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, 233846052, 2572306572, 28295372292, 311249095212, 3423740047332, 37661140520652, 414272545727172, 4556998002998892, 50126978032987812, 551396758362865932, 6065364341991525252, 66719007761906777772, 733909085380974555492

Offset: 0

N. J. A. Sloane

Coordination sequence for infinite tree with valency 12.
The n-th term of the coordination sequence of the infinite tree with valency 2m is the same as the number of reduced words of size n in the free group on m generators. In the five sequences A003946, A003948, A003950, A003952, A003954 m is 2, 3, 4, 5, 6 . - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001 and Ola Veshta (olaveshta(AT)my-deja.com), Mar 30 2001.
For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,...,11} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..900
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 313.
Index to divisibility sequences.
Index entries for linear recurrences with constant coefficients, signature (11).
Index entries for sequences related to trees.

Cf. A003946, A003948, A003950, A003952, A003953, A003954, A029653.

Concatenation([1], List([1..20], n-> 12*11^(n-1) )); # G. C. Greubel, Sep 23 2019

[1] cat [12*11^(n-1): n in [1..20]]; // Vincenzo Librandi, Dec 11 2012

k:=12; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # modified by G. C. Greubel, Sep 24 2019

Join[{1}, 12*11^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
CoefficientList[Series[(1+x)/(1-11x), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 10 2012 *)

a(n)=12*11^n\11 \\ Charles R Greathouse IV, Aug 14 2015

[1]+[12*11^(n-1) for n in (1..20)] # G. C. Greubel, Sep 23 2019

a(n) = Sum_{k=0..n} A029653(n,k)*x^k for x = 10. - Philippe Deléham, Jul 10 2005
G.f.: (1+x)/(1-11*x). The Hankel transform of this sequence is [1,-12,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007
a(0) = 1; for n>0, a(n) = 12*11^(n-1). - Vincenzo Librandi, Nov 18 2010
a(0) = 1, a(1)=12, a(n) = 11*a(n-1). - Vincenzo Librandi, Dec 10 2012
E.g.f.: (12*exp(11*x) - 1)/11. - Elmo R. Oliveira, Mar 24 2025

Edited by N. J. A. Sloane, Dec 04 2009

A170732 Expansion of g.f.: (1+x)/(1 - 12*x).

Original entry on oeis.org

1, 13, 156, 1872, 22464, 269568, 3234816, 38817792, 465813504, 5589762048, 67077144576, 804925734912, 9659108818944, 115909305827328, 1390911669927936, 16690940039135232, 200291280469622784, 2403495365635473408, 28841944387625680896, 346103332651508170752

Offset: 0

N. J. A. Sloane, Dec 05 2009

For n >= 1, a(n) equals the number of words of length n-1 on the alphabet {0,1,...,12} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015

Kenny Lau, Table of n, a(n) for n = 0..926
M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, J. Int. Seq. 18 (2015) # 15.4.7.
Index entries for linear recurrences with constant coefficients, signature (12).

Cf. A003945, A003953, A003954.

k:=13;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Sep 24 2019

k:=13; [1] cat [k*(k-1)^(n-1): n in [1..25]]; // G. C. Greubel, Sep 24 2019

k:=13; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # G. C. Greubel, Sep 24 2019

Join[{1}, 13*12^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
Join[{1},NestList[12#&,13,20]] (* Harvey P. Dale, Nov 24 2024 *)

a(n)=if(n,13*12^(n-1),1) \\ Charles R Greathouse IV, Jul 01 2016

for i in range(1001):print(i,13*12**(i-1) if i>0 else 1) # Kenny Lau, Aug 01 2017

k=13; [1]+[k*(k-1)^(n-1) for n in (1..25)] # G. C. Greubel, Sep 24 2019

a(0)=1; for n > 0, a(n) = 13*12^(n-1). - Vincenzo Librandi, Dec 05 2009

E.g.f.: (13*exp(12*x) - 1)/12. - G. C. Greubel, Sep 24 2019

A042950 Row sums of the Lucas triangle A029635.

Original entry on oeis.org

2, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944

Offset: 0

N. J. A. Sloane and J. H. Conway

Map a binary sequence b=[ b_1,...] to a binary sequence c=[ c_1,...] so that C = 1/Product((1-x^i)^c_i == 1 + Sum b_i*x^i mod 2.
This produces 2 new sequences: d={i:c_i=1} and e=[ 1,e_1,... ] where C = 1 + Sum e_i*x^i.
This sequence is d when b=[ 0,1,1,1,1,...].
Number of rises after n+1 iterations of morphism A007413.
a(n) written in base 2: a(0) = 10, a(n) for n >= 1: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, (n-1) times 0 (see A003953(n)). - Jaroslav Krizek, Aug 17 2009
Row sums of the Lucas triangle A029635. - Sergio Falcon, Mar 17 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Kitaev and T. Mansour, Counting the occurrences of generalized patterns in words generated by a morphism, arXiv:math/0210170 [math.CO], 2002.
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A003945, A007283.

[2] cat [2^(n+1) - 2^(n-1): n in [1..40]]; // Vincenzo Librandi, Aug 08 2015

Table[ Ceiling[3*2^(n - 1)], {n, 0, 32}] (* Robert G. Wilson v, Jul 08 2006 *)
a[0] = 2; a[1] = 3; a[n_] := 2a[n - 1]; Table[a[n], {n, 0, 32}] (* Robert G. Wilson v, Jul 08 2006 *)
f[s_] := Append[s, 1 + Plus @@ s]; Nest[f, {2}, 32] (* Robert G. Wilson v, Jul 08 2006 *)
CoefficientList[Series[(2 - x)/(1 - 2x), {x, 0, 32}], x] (* Robert G. Wilson v, Jul 08 2006 *)

PARI
```
a(n)=ceil(3*2^(n-1))
```

def A042950(n): return (3*2**n + int(n==0))//2 # G. C. Greubel, Jun 06 2025

G.f.: (2-x)/(1-2*x).
a(n) = 2*a(n-1), n > 1; a(0)=2, a(1)=3.
a(n) = A003945(n), for n > 0.
From Paul Barry, Dec 06 2004: (Start)
Binomial transform of 2, 1, 2, 1, 2, 1, ... = (3+(-1)^n)/2.
a(n) = (3*2^n + 0^n)/2. (End)
a(0) = 2, a(n) = 3*2^(n-1) = 2^n + 2^(n-1) for n >= 1. - Jaroslav Krizek, Aug 17 2009
a(n) = 2^(n+1) - 2^(n-1), for n > 0. - Ilya Gutkovskiy, Aug 08 2015
E.g.f.: (3*exp(2*x) + 1)/2. - G. C. Greubel, Jun 06 2025

A098011 10^a(n) + 1 = A088773(n).

Original entry on oeis.org

1, 1, 2, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944, 12884901888

Offset: 1

Ray G. Opao, Sep 09 2004

Starting from the 4th term, every succeeding term is twice the preceding term. I.e., a(n+1) = 2a(n).
Number of binary words of length n-2 that do not start with 01 (n>=2). Example: a(5)=6 because we have 000,001,100,101,110 and 111. Except for the initial term, column 0 of A119440. - Emeric Deutsch, May 19 2006
a(n) written in base 2: a(1) = 1, a(2) = 1, a(3) = 10, a(n) for n >= 4: 11, 110, 1100, 11000, 110000, ..., i.e.: 2 times 1, (n-4) times 0 (see A003953(n-3)). - Jaroslav Krizek, Aug 17 2009
a(n) for n > 1 are the values used in the variant of the game 2048 called "threes". - Michael De Vlieger, Jul 18 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
David Eppstein, Making Change in 2048, arXiv:1804.07396 [cs.DM], 2018.
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A119440.

a:=proc(n) if n=1 or n=2 then 1 elif n=3 then 2 else 3*2^(n-4) fi end: seq(a(n),n=1..37); # Emeric Deutsch, May 19 2006

Table[ Ceiling[3*2^(n - 4)], {n, 34}] (* or *)
Rest@CoefficientList[Series[x(1 - x - x^3)/(1 - 2x), {x, 0, 33}], x] (* Robert G. Wilson v, Jul 08 2006 *)
Table[Ceiling[2^{n-2}]-Floor[2^{n-4}],{n,1,10}] (* Martin Grymel, Oct 17 2012 *)

x='x+O('x^99); Vec(x*(1-x-x^3)/(1-2*x)) \\ Altug Alkan, Jul 18 2018

G.f.: x*(1 - x - x^3)/(1 - 2*x). - Paul Barry, Feb 17 2005
a(n) = 3*2^(n-4) for n>3; a(1)=a(2)=1, a(3)=2. - Emeric Deutsch, May 19 2006
a(n) = 2^(n-4) + 2^(n-3) for n > 3. - Jaroslav Krizek, Aug 17 2009
a(1) = 1, a(2) = 1, a(3) = 2, for n > 3: a(n) = Sum_{i = 2..n-1} a(i). - Jaroslav Krizek, Nov 16 2009 [Corrected by Petros Hadjicostas, Nov 16 2019]
a(n) = A042950(n-3). - Philippe Deléham, Oct 17 2011
a(n) = ceiling(2^{n-2}) - floor(2^{n-4}). - Martin Grymel, Oct 17 2012

More terms from Emeric Deutsch, May 19 2006

A167914 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

Original entry on oeis.org

1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, 110000000, 1100000000, 11000000000, 110000000000, 1100000000000, 11000000000000, 110000000000000, 1100000000000000, 10999999999999945, 109999999999998900

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

nonn

The initial terms coincide with those of A003953, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,-45).

Cf. A003953, A154638, A169452.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+x)*(1-x^16)/(1-10*x+54*x^16-45*x^17) )); // G. C. Greubel, Dec 04 2024

CoefficientList[Series[(1+t)*(1-t^16)/(1-10*t+54*t^16-45*t^17), {t,0,50}], t] (* G. C. Greubel, Jul 01 2016; Dec 04 2024 *)
coxG[{16,45,-9}] (* The coxG program is at A169452 *) (* G. C. Greubel, Dec 04 2024 *)

def A167914_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-10*x+54*x^16-45*x^17) ).list()
A167914_list(40) # G. C. Greubel, Dec 04 2024

G.f.: (t^16 + 2*t^15 + 2*t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/( 45*t^16 - 9*t^15 - 9*t^14 - 9*t^13 - 9*t^12 - 9*t^11 - 9*t^10 - 9*t^9 - 9*t^8 - 9*t^7 - 9*t^6 - 9*t^5 - 9*t^4 - 9*t^3 - 9*t^2 - 9*t + 1).
From G. C. Greubel, Dec 04 2024: (Start)
a(n) = 9*Sum_{j=1..15} a(n-j) - 45*a(n-16).
G.f.: (1+x)*(1-x^16)/(1 - 10*x + 54*x^16 - 45*x^17). (End)

A122391 Dimension of 2-variable non-commutative harmonics (Hausdorff derivative). The dimension of the space of non-commutative polynomials in 2 variables which are killed by all symmetric differential operators (where for a monomial w, d_{xi} ( w ) = sum over all subwords of w deleting xi once).

Original entry on oeis.org

1, 1, 1, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944, 12884901888, 25769803776

Offset: 0

Mike Zabrocki, Aug 31 2006

Except for first couple of terms, series agrees with A003945.
a(n) written in base 2: a(0) = 1, a(1) = 1, a(2) = 1, a(n) for n >= 3: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, (n-3) times 0 (see A003953(n-2)). - Jaroslav Krizek, Aug 17 2009
For n>=2, a(n) equals the numbers of words of length n-2 on alphabet {0,1,2} containing no subwords 00, 11 and 22. - Milan Janjic, Jan 31 2015
Also the number of compositions of n whose first or last part is equal to 1, for n >= 1. - Peter Luschny, Jan 29 2024

			a(1) = 1 because x1 - x2 is killed by d_x1 + d_x2.
a(2) = 1 because x1 x2 - x2 x1 is killed by d_x1+d_x2, d_x1^2 + d_x2^2.
a(3) = 3 because x1 x1 x2 - 2 x1 x2 x1 + x2 x1 x1, x1 x2 x2 - 2 x2 x1 x2 + x2 x2 x1, x1 x1 x2 - x1 x2 x1 - x2 x1 x2 + x2 x2 x1 are all killed by d_x1 + d_x2, d_x1^2 + d_x2^2, d_x1 d_x2, d_x1^3 + d_x2^3 and d_x1^2 d_x2 + d_x1 d_x2^2.
From _Peter Luschny_, Jan 29 2024: (Start)
Compositions of n with 1 in the first or the last slot.
 1: [1];
 2: [1, 1];
 3: [1, 1, 1], [1, 2], [2, 1];
 4: [1, 1, 1, 1], [1, 1, 2], [1, 2, 1], [1, 3], [2, 1, 1], [3, 1];
 5: [1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 1, 2, 1], [1, 1, 3], [1, 2, 1, 1], [1, 2, 2], [1, 3, 1], [1, 4], [2, 1, 1, 1], [2, 2, 1], [3, 1, 1], [4, 1].
(End)

C. Reutenauer, Free Lie algebras. London Mathematical Society Monographs. New Series, 7. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. xviii+269 pp.

N. Bergeron, C. Reutenauer, M. Rosas, and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math/0502082 [math.CO], 2005. See also, Canad. J. Math. 60 (2008), no. 2, 266-296.
C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A003945, A034008, A122367, A122392, A122393, A122394.

coeffs(convert(series((1-q)*(1-q^2)/(1-2*q),q,20),`+`)-O(q^20),q);

Table[Ceiling[2^(n-2)] + Floor[2^(n-3)], {n,0,30}] (* Martin Grymel, Oct 17 2012 *)

G.f.: (1-q)*(1-q^2)/(1-2*q).
a(n) = 2^n - 2^(n-1) - 2^(n-2) + 2^(n-3) (for n > 2).
a(0) = 1, a(1) = 1, a(2) = 1, a(n) = 3*2^(n-3) for n > 2.
a(n) = 3*2^(n-3) = 2^(n-3) + 2^(n-2) for n >= 3. - Jaroslav Krizek, Aug 17 2009
a(n) = ceiling(2^(n-2)) + floor(2^(n-3)). - Martin Grymel, Oct 17 2012
E.g.f.: (5 + 3*exp(2*x) + 2*x - 2*x^2)/8. - Stefano Spezia, Jan 26 2025

More terms from Michel Marcus, Jan 26 2025

A266180 Decimal representation of the n-th iteration of the "Rule 6" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 6, 16, 96, 256, 1536, 4096, 24576, 65536, 393216, 1048576, 6291456, 16777216, 100663296, 268435456, 1610612736, 4294967296, 25769803776, 68719476736, 412316860416, 1099511627776, 6597069766656, 17592186044416, 105553116266496, 281474976710656

Offset: 0

Robert Price, Dec 22 2015

A001025 is a subsequence. - Altug Alkan, Dec 23 2015

Rules 38, 134 and 166 also generate this sequence.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..999
Hans Montanus and Ron Westdijk, Cellular Automation and Binomials, Green Blue Mathematics (2022), p. 22.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (0,16).

Cf. A001025, A266178, A266179, A019590, A003953, A003945, A000034, A032766, A042948, A035608.

rule=6; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)
LinearRecurrence[{0,16},{1,6},30] (* Harvey P. Dale, May 25 2016 *)

print([int(4**(n-1)*(5-(-1)**n)) for n in range(30)]) # Karl V. Keller, Jr., Jun 03 2021

From Colin Barker, Dec 23 2015 and Apr 13 2019: (Start)
a(n) = 4^(n-1)*(5-(-1)^n).
a(n) = 16*a(n-2) for n>1.
G.f.: (1+6*x) / ((1-4*x)*(1+4*x)).
(End)

A292680 Rule 6: 000, ..., 111 -> 0, 1, 1, 0, 0, 0, 0, 0.

Original entry on oeis.org

0, 6, 12, 8, 24, 26, 16, 16, 48, 54, 52, 48, 32, 34, 32, 32, 96, 102, 108, 104, 104, 106, 96, 96, 64, 70, 68, 64, 64, 66, 64, 64, 192, 198, 204, 200, 216, 218, 208, 208, 208, 214, 212, 208, 192, 194, 192, 192, 128, 134, 140, 136, 136, 138, 128, 128, 128, 134, 132, 128

Offset: 0

M. F. Hasler, Oct 09 2017

The orbit of 1 under this rule is A266180.
Rule 6 is the smallest rule which is even (otherwise infinitely many bits would be switched on at step 1, for any finite starting value) and nontrivial (i.e., does not lead to extinction nor simple reproduction, possibly shifted left or right, of a single-bit initial state).
As is customary in the context of elementary cellular automata, the result is the bitmap obtained from the argument extended by one bit to the right (as to consider the cell which has bit 0 of the input as left neighbor), cf. example. Since the rule has a value < 16, no cell having its left neighbor 'on' will be on. Therefore all values a(n) are even. See A292681 for the variant without this extension beyond bit 0, i.e., a(n)/2.

			     n        |          a(n)
   0 =   0[2] |       0[2] =  0
   1 =   1[2] |     110[2] =  6 (bits below 001 and 010 are on, below 100 is off)
   2 =  10[2] |    1100[2] = 12 (as above, plus an additional bit 0 below 000)
   3 =  11[2] |    1000[2] =  8 (1 below 001, 0 below 011, 110 and 100.)
   4 = 100[2] |   11000[2] = 24 (as n = 1 and n = 2, shifted right once more)
   5 = 101[2] |   11010[2] = 26 (1 below 001 and 010 (twice), 0 below 101 and 100)
   6 = 110[2] |   10000[2] = 16 (as n = 3, shifted right once)
   7 = 111[2] |   10000[2] = 16 (1 below 001, 0 below 011, 111, 110 and 100).

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A266178, A266179, A266180, A019590, A003953, A003945, A000034, A032766, A042948, A035608.

apply( A292680(n,r=6)=sum(i=0,logint(!n+n<<=2,2)+1,bittest(r,(n>>i)%8)<

cp's OEIS Frontend

A007283 a(n) = 3*2^n.

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A003945 Expansion of g.f. (1+x)/(1-2*x).

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A003954 Expansion of g.f.: (1+x)/(1-11*x).

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A170732 Expansion of g.f.: (1+x)/(1 - 12*x).

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A042950 Row sums of the Lucas triangle A029635.

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A098011 10^a(n) + 1 = A088773(n).

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