A008776 -id:A008776 - cp's OEIS frontend

A001021 Powers of 12.

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

A048473 a(0)=1, a(n) = 3a(n-1) + 2; a(n) = 23^n - 1.

Original entry on oeis.org

1, 5, 17, 53, 161, 485, 1457, 4373, 13121, 39365, 118097, 354293, 1062881, 3188645, 9565937, 28697813, 86093441, 258280325, 774840977, 2324522933, 6973568801, 20920706405, 62762119217, 188286357653, 564859072961, 1694577218885, 5083731656657, 15251194969973, 45753584909921

Offset: 0

Clark Kimberling

The number of triangles (of all sizes, including holes) in Sierpiński's triangle after n inscriptions. - Lee Reeves, May 10 2004
The sequence is not only related to Sierpiński's triangle, but also to "Floret's cube" and the quaternion factor space Q X Q / {(1,1), (-1,-1)}. It can be written as a_n = ves((A+1)x)^n) as described at the Math Forum Discussions link. - Creighton Dement, Jul 28 2004
Relation to C(n) = Collatz function iteration using only odd steps: If we look for record subsequences where C(n) > n, this subsequence starts at 2^n - 1 and stops at the local maximum of 2*3^n - 1. Examples: [3,5], [7,11,17], [15,23,35,53], ..., [127,191,287,431,647,971,1457]. - Lambert Klasen, Mar 11 2005
Group the natural numbers so that the (2n-1)-th group sum is a multiple of the (2n)-th group containing one term. (1,2),(3),(4,5,6,7,8,9,10,11),(12),(13,14,15,16,17,18,19,...,38),(39),(40,41,...,118,119),(120), (121,122,123,...) ... a(n) = {the sum of the terms of (2n-1)-th group}/{the term of (2n)th group}. The first term of the odd numbered group is given by A003462. The only term of even numbered group is given by A029858. - Amarnath Murthy, Aug 01 2005
a(n)+1 = A008776(n); it appears that this gives the number of terms in the (n+1)-th "gap" of numbers missing in A171884. - M. F. Hasler, May 09 2013
Sum of n-th row of triangle of powers of 3: 1; 1 3 1; 1 3 9 3 1; 1 3 9 27 9 3 1; ... - Philippe Deléham, Feb 23 2014
For n >= 3, also the number of dominating sets in the n-helm graph. - Eric W. Weisstein, May 28 2017
The number of elements of length <= n in the free group on two generators. - Anton Mellit, Aug 10 2017
In general, a first order inhomogeneous recurrence of the form s(0) = a, s(n) = m*s(n-1) + k, n>0, will have a closed form of a*m^n + ((m^n-1)/(m-1))*k. - Gary Detlefs, Jun 07 2024

			a(0) = 1;
a(1) = 1 + 3 + 1 = 5;
a(2) = 1 + 3 + 9 + 3 + 1 = 17;
a(3) = 1 + 3 + 9 + 27 + 9 + 3 + 1 = 53; etc. - _Philippe Deléham_, Feb 23 2014

Theoni Pappas, Math Stuff, Wide World Publ/Tetra, San Carlos CA, page 15, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Creighton Dement, A paper on floretions and quaternions, some questions, The Math Forum.
Eric Weisstein's World of Mathematics, Dominating Set.
Eric Weisstein's World of Mathematics, Helm Graph.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

a(n)=T(2,n), array T given by A048471.
Cf. A003462, A029858. A column of A119725.
Cf. A008776, A010684, A046055, A112468, A112739, A134347, A171884.

[2*3^n - 1: n in [0..30]]; // Vincenzo Librandi, Sep 23 2011

g:= ((1+x)/(1-3*x)/(1-x)): gser:=series(g, x=0, 43): seq(coeff(gser, x, n), n=0..30); # Zerinvary Lajos, Jan 11 2009; typo fixed by Marko Mihaily, Mar 07 2009

NestList[3 # + 2 &, 1, 30] (* Harvey P. Dale, Mar 06 2012 *)
LinearRecurrence[{4, -3}, {1, 5}, 30] (* Harvey P. Dale, Mar 06 2012 *)
Table[2 3^n - 1, {n, 20}] (* Eric W. Weisstein, May 28 2017 *)
2 3^Range[20] - 1 (* Eric W. Weisstein, May 28 2017 *)

first(m)=vector(m,n,n--;2*3^n - 1) \\ Anders Hellström, Dec 11 2015

n-th difference of a(n), a(n-1), ..., a(0) is 2^(n+1) for n=1, 2, 3, ...
a(0)=1, a(n) = a(n-1) + 3^n + 3^(n-1). - Lee Reeves, May 10 2004
a(n) = (3^n + 3^(n+1) - 2)/2. - Creighton Dement, Jul 31 2004
(1, 5, 17, 53, 161, ...) = Ternary (1, 12, 122, 1222, 12222, ...). - Gary W. Adamson, May 02 2005
Row sums of triangle A134347. Also, binomial transform of A046055: (1, 4, 8, 16, 32, 64, ...); and double binomial transform of A010684: (1, 3, 1, 3, 1, 3, ...). - Gary W. Adamson, Oct 21 2007
G.f.: (1+x)/((1-3*x)*(1-x)). - Zerinvary Lajos, Jan 11 2009; corrected by R. J. Mathar, Jan 21 2009
a(0)=1, a(1)=5, a(n) = 4*a(n-1) - 3*a(n-2). - Harvey P. Dale, Mar 06 2012
a(n) = Sum_{k=0..n} A112468(n,k)*4^k. - Philippe Deléham, Feb 23 2014
E.g.f.: exp(x)*(2*exp(2*x) - 1). - Elmo R. Oliveira, Mar 08 2025

Better description from Amarnath Murthy, May 27 2001

A057711 a(0)=0, a(1)=1, a(n) = n*2^(n-2) for n >= 2.

Original entry on oeis.org

0, 1, 2, 6, 16, 40, 96, 224, 512, 1152, 2560, 5632, 12288, 26624, 57344, 122880, 262144, 557056, 1179648, 2490368, 5242880, 11010048, 23068672, 48234496, 100663296, 209715200, 436207616, 905969664, 1879048192, 3892314112, 8053063680

Offset: 0

Bernhard Wolf (wolf(AT)cs.tu-berlin.de), Oct 24 2000

Number of states in the planning domain FERRY, when n-3 cars are at one of two shores while the (n-2)nd car may be on the ferry or at one of the shores.
If the ferry could board any number of cars (instead of only one), the number of states would form the Pisot sequence P(2,6) (A008776). In addition, if k shores existed, the sequence would form the Pisot sequence P(k,k(k+1)). This corresponds to the BRIEFCASE planning domain.
a(i) is the number of occurrences of the number 1 in all palindromic compositions of n = 2*(i+1). - Silvia Heubach (sheubac(AT)calstatela.edu), Jan 10 2003. E.g., there are 5 palindromic compositions of 6, namely 111111 11211 2112 1221 141, containing a total of 16 1's.
Number of occurrences of 00's in all circular binary words of length n. Example: a(3)=6 because in the circular binary words 000, 001, 010, 011, 100, 101, 110 and 111 we have a total of 3+1+1+0+1+0+0+0=6 occurrences of 00. a(n) = Sum_{k=0..n} k*A119458(n,k). - Emeric Deutsch, May 20 2006
a(n) is the number of permutations on [n] for which the entries of each left factor form a circular subinterval of [n]. A subset I of [n] forms a circular subinterval of [n] if it is an ordinary interval [a,b] or has the form [1,a]-union-[b,n] for 1 <= a < b <= n. For example, (5,4,2) is a left factor of the permutation (5,4,2,1,3) which does not form a circular subinterval of [5] and a(4)=16 counts all 24 permutations of [4] except the eight whose first two entries are 1,3 (in either order) or 2,4. - David Callan, Mar 30 2007
a(n) is the total number of runs in all Boolean (n-1)-strings. For example, the 8 Boolean 3-strings, 000, 001, 010, 011, 100, 101, 110, 111 have 1, 2, 3, 2, 2, 3, 2, 1 runs respectively. - David Callan, Jul 22 2008
From Gary W. Adamson, Jul 31 2010: (Start)
Starting with "1" = (1, 2, 4, 8, ...) convolved with (1, 0, 2, 4, 8, ...).
Example: a(6) = 96 = (32, 16, 8, 4, 2, 1) dot (1, 0, 2, 4, 8, 16) = (32 + 0 + 16 + 16 + 16, + 16) = 32 + 4*16 (End)
An elephant sequence, see A175654. For the corner squares 24 A[5] vectors, with decimal values between 27 and 432, lead to this sequence (without the leading 0). For the central square these vectors lead to the companion sequence A087447 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010
Starting with 1 = (1, 1, 2, 4, 8, 16, ...) convolved with (1, 1, 3, 7, 15, 31, ...). - Gary W. Adamson, Oct 26 2010
a(n) is the number of ways to draw simple polygonal chains for n vertices lying on a circle. - Anton Zakharov, Dec 31 2016
Also the number of edges, maximal cliques, and maximum cliques in the n-folded cube graph for n > 3. - Eric W. Weisstein, Dec 01 2017 and Mar 21 2018
Number of pairs of compositions of n corresponding to a seaweed algebra of index n-2 for n > 2. - Nick Mayers, Jun 25 2018
Starting with 1, 2, 6, 16, ..., number of permutations of length n>0 avoiding the partially ordered pattern (POP) {1>2, 1>3} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the second and third elements. - Sergey Kitaev, Dec 08 2020

			a(1)=6 because the palindromic compositions of n=4 are 4, 1+2+1, 1+1+1+1 and 2+2 and they contain 6 ones. - Silvia Heubach (sheubac(AT)calstatela.edu), Jan 10 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
O. Aichholzer, A. Asinowski, and T. Miltzow, Disjoint compatibility graph of non-crossing matchings of points in convex position, arXiv preprint arXiv:1403.5546 [math.CO], 2014.
A. Burstein, S. Kitaev, and T. Mansour, Partially ordered patterns and their combinatorial interpretations, PU. M. A. Vol. 19 (2008), No. 2-3, pp. 27-38.
P. Chinn, R. Grimaldi and S. Heubach, The frequency of summands of a particular size in Palindromic Compositions, Ars Combin. 69 (2003), 65-78.
Vincent Coll et al., Meander graphs and Frobenius seaweed Lie algebras II, Journal of Generalized Lie Theory and Applications 9.1 (2015).
Vladimir Dergachev and Alexandre Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10.2 (2000): 331-343.
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.
M. Ghallab et al., FERRY domain
M. Ghallab, A. Howe et al., PDDL - The Planning Domain Definition Language, Version 1.2, Technical Report CVC TR-98-003/DCS TR-1165. Yale Center for Computational Vision and Control, 1998.
Anna Khmelnitskaya, Gerard van der Laan, and Dolf Talmanm, The Number of Ways to Construct a Connected Graph: A Graph-Based Generalization of the Binomial Coefficients, J. Int. Seq. (2023) Art. 23.4.3. See p. 12.
Eric Weisstein's World of Mathematics, Folded Cube Graph
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Maximum Clique
B. Wolf, Creating state sets
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A082133, A082134, A082135, A082136.
Pisot sequence P(2, 6) (A008776), Pisot sequence P(k, k(k+1))
Cf. A119458.
Cf. A082140, A082141, A082138, A082139, A080951, A080929.

[Ceiling(n*2^(n-2)) : n in [0..40]]; // Vincenzo Librandi, Sep 22 2011

Join[{0, 1}, Table[n 2^(n - 2), {n, 2, 30}]] (* Eric W. Weisstein, Dec 01 2017 *)
Join[{0, 1}, LinearRecurrence[{4, -4}, {2, 6}, 20]] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[x (1 - 2 x + 2 x^2)/(1 - 2 x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)

a(n)=ceil(n*2^(n-2)) \\ Charles R Greathouse IV, Oct 31 2011

x='x+O('x^50); concat(0, Vec(x*(1-2*x+2*x^2)/(1-2*x)^2)) \\ Altug Alkan, Nov 01 2015

a(n) = ceiling(n*2^(n-2)).
Binomial transform of (0, 1, 0, 3, 0, 5, 0, 7, ...).
From Paul Barry, Apr 06 2003: (Start)
a(0)=0, a(n) = n*(0^(n-1) + 2^(n-1))/2, n > 0.
a(n) = Sum_{k=0..n} binomial(n, 2k+1)*(2k+1).
E.g.f.: x*exp(x)*cosh(x). (End)
The sequence 1, 1, 6, 16, ... is the binomial transform of A016813 with interpolated zeros. - Paul Barry, Jul 25 2003
For n > 1, a(n) = Sum_{k=0..n} (k-n/2)^2 C(n, k). (n+1)*a(n) = A001788(n). - Mario Catalani (mario.catalani(AT)unito.it), Nov 26 2003
From Paul Barry, May 07 2004: (Start)
a(n) = n*2^(n-2) - Sum_{k=0..n} binomial(n, k)*k*(-1)^k.
G.f.: x*(1-2*x+2*x^2)/(1-2*x)^2. (End)
a(n+1) = ceiling(binomial(n+1,1)*2^(n-1)). - Zerinvary Lajos, Nov 01 2006
a(n+1) = Sum_{k=0..n} A196389(n,k)*2^k. - Philippe Deléham, Oct 31 2011
a(0)=0, a(1)=1, a(2)=2, a(3)=6, a(n+1) = 4*a(n)-4*a(n-1) for n >= 3. - Philippe Deléham, Feb 20 2013
a(n) = A002064(n-1) - A002064(n-2), for n >= 2. - Ivan N. Ianakiev, Dec 29 2013
From Amiram Eldar, Aug 05 2020: (Start)
Sum_{n>=1} 1/a(n) = 4*log(2) - 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(3/2) - 1. (End)

A009975 Powers of 31: a(n) = 31^n.

Original entry on oeis.org

1, 31, 961, 29791, 923521, 28629151, 887503681, 27512614111, 852891037441, 26439622160671, 819628286980801, 25408476896404831, 787662783788549761, 24417546297445042591, 756943935220796320321, 23465261991844685929951, 727423121747185263828481

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 31), L(1, 31), P(1, 31), T(1, 31). Essentially same as Pisot sequences E(31, 961), L(31, 961), P(31, 961), T(31, 961). See A008776 for definitions of Pisot sequences.

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 31-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

T. D. Noe, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (31).

Table[31^n, {n, 0, 15}] (* Robert P. P. McKone, Jan 04 2022 *)

a(n)=31^n \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 1/(1-31*x). - Philippe Deléham, Nov 24 2008
E.g.f.: exp(31x). - Geoffrey Critzer, Feb 28 2009
a(n) = 31*a(n-1). - Zerinvary Lajos, Apr 29 2009

A009973 Powers of 29.

Original entry on oeis.org

1, 29, 841, 24389, 707281, 20511149, 594823321, 17249876309, 500246412961, 14507145975869, 420707233300201, 12200509765705829, 353814783205469041, 10260628712958602189, 297558232675799463481, 8629188747598184440949

Offset: 0

N. J. A. Sloane, Dec 11 1996

Same as Pisot sequences E(1, 29), L(1, 29), P(1, 29), T(1, 29). Essentially same as Pisot sequences E(29, 841), L(29, 841), P(29, 841), T(29, 841). See A008776 for definitions of Pisot sequences.
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 29-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Numbers n such that sigma(29*n) = 29*n + sigma(n). - Jahangeer Kholdi, Nov 23 2013

T. D. Noe, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (29).

[29^n: n in [0..30]]; // Vincenzo Librandi, Nov 21 2010

A009973:=n->29^n; seq(A009973(n), n=0..20); # Wesley Ivan Hurt, Jun 06 2014

29^Range[0,20] (* Harvey P. Dale, Jul 19 2011 *)

makelist(29^n, n, 0, 30); /* Martin Ettl, Nov 07 2012 */

a(n)=29^n \\ Charles R Greathouse IV, Sep 24 2015

[29^n for n in (1..17)] # Zerinvary Lajos, Apr 29 2009

G.f.: 1/(1-29*x). - Philippe Deléham, Nov 24 2008
a(n) = 29^n; a(n) = 29*a(n-1) for n>0, a(0)=1. - Vincenzo Librandi, Nov 21 2010
G.f.: (G(0)-2)/(9*x), where G(k)= 1 + 1/(1 - x*(9*k-9)/(x*(9*k+20) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013

A009964 Powers of 20.

Original entry on oeis.org

1, 20, 400, 8000, 160000, 3200000, 64000000, 1280000000, 25600000000, 512000000000, 10240000000000, 204800000000000, 4096000000000000, 81920000000000000, 1638400000000000000, 32768000000000000000

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 20), L(1, 20), P(1, 20), T(1, 20). Essentially same as Pisot sequences E(20, 400), L(20, 400), P(20, 400), T(20, 400). See A008776 for definitions of Pisot sequences.
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 20-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
a(n) gives the number of small cubes in the n-th iteration of the Menger sponge fractal. - Felix Fröhlich, Jul 09 2016
Equivalently, the number of vertices in the n-Menger sponge graph.

T. D. Noe, Table of n, a(n) for n = 0..100
Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Menger Sponge
Eric Weisstein's World of Mathematics, Menger Sponge Graph
Eric Weisstein's World of Mathematics, Vertex Count
Wikipedia, Menger sponge
Index entries for linear recurrences with constant coefficients, signature (20).

Cf. A291066 (edge count).
Cf. A000079, A011557; A000302, A000351; A000244, A159991.
Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.

List([0..20],n->20^n); # Muniru A Asiru, Nov 21 2018

[20^n: n in [0..100]] // Vincenzo Librandi, Nov 21 2010

[20^n$n=0..20]; # Muniru A Asiru, Nov 21 2018

20^Range[0, 10] (* or *) LinearRecurrence[{20}, {1}, 20] (* Eric W. Weisstein, Aug 17 2017 *)

makelist(20^n,n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=20^n \\ Charles R Greathouse IV, Jun 19 2015

powers(20,12) \\ Charles R Greathouse IV, Jun 19 2015

[20**n for n in range(21)] # Stefano Spezia, Nov 21 2018

[20^n for n in range(21)] # Zerinvary Lajos, Apr 29 2009

G.f.: 1/(1-20*x).
E.g.f.: exp(20*x).
a(n) = A159991(n)/A000244(n). - Reinhard Zumkeller, May 02 2009
From Vincenzo Librandi, Nov 21 2010: (Start)
a(n) = 20^n.
a(n) = 20*a(n-1) for n > 0, a(0) = 1. (End)
a(n) = A000079(n)*A011557(n) = A000302(n)*A000351(n). - Felix Fröhlich, Jul 09 2016

A001029 Powers of 19.

Original entry on oeis.org

1, 19, 361, 6859, 130321, 2476099, 47045881, 893871739, 16983563041, 322687697779, 6131066257801, 116490258898219, 2213314919066161, 42052983462257059, 799006685782884121, 15181127029874798299, 288441413567621167681, 5480386857784802185939, 104127350297911241532841, 1978419655660313589123979, 37589973457545958193355601

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 19), L(1, 19), P(1, 19), T(1, 19). Essentially same as Pisot sequences E(19, 361), L(19, 361), P(19, 361), T(19, 361). See A008776 for definitions of Pisot sequences.

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 19-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 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 283
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 (19).

[ 19^n: n in [0..20] ]; // Vincenzo Librandi, Nov 21 2010

[ n eq 1 select 1 else 19*Self(n-1): n in [1..21] ];

A001029:=-1/(-1+19*z); # conjectured by Simon Plouffe in his 1992 dissertation

Table[19^n,{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

a(n)=19^n \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 1/(1-19x), e.g.f.: exp(19x)

a(n) = 19^n; a(n) = 19*a(n-1) with a(0)=1. - Vincenzo Librandi, Nov 21 2010

A009967 Powers of 23.

Original entry on oeis.org

1, 23, 529, 12167, 279841, 6436343, 148035889, 3404825447, 78310985281, 1801152661463, 41426511213649, 952809757913927, 21914624432020321, 504036361936467383, 11592836324538749809, 266635235464391245607, 6132610415680998648961, 141050039560662968926103, 3244150909895248285300369, 74615470927590710561908487, 1716155831334586342923895201

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 23), L(1, 23), P(1, 23), T(1, 23). Essentially same as Pisot sequences E(23, 529), L(23, 529), P(23, 529), T(23, 529). See A008776 for definitions of Pisot sequences.
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 23-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Numbers k such that sigma(23*k) = 23*k + sigma(k). - Jahangeer Kholdi, Nov 23 2013

T. D. Noe, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (23).

Cf. A000079, A008776, A009990.

[23^n: n in [0..100]]; // Vincenzo Librandi, Nov 21 2010

23^Range[0,20]  (* Harvey P. Dale, Apr 04 2011 *)

A009967(n):=23^n$
makelist(A009967(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

a(n)=23^n \\ Charles R Greathouse IV, Sep 24 2015

[lucas_number1(n,23,0) for n in range(1, 17)] # Zerinvary Lajos, Apr 29 2009

G.f.: 1/(1-23*x). - Philippe Deléham, Nov 23 2008
a(n) = 23^n; a(n) = 23*a(n-1) n>0 a(0)=1. - Vincenzo Librandi, Nov 21 2010
From Elmo R. Oliveira, Jul 10 2025: (Start)
E.g.f.: exp(23*x).
a(n) = A009990(n)/A000079(n). (End)

A001024 Powers of 15.

Original entry on oeis.org

1, 15, 225, 3375, 50625, 759375, 11390625, 170859375, 2562890625, 38443359375, 576650390625, 8649755859375, 129746337890625, 1946195068359375, 29192926025390625, 437893890380859375, 6568408355712890625, 98526125335693359375, 1477891880035400390625, 22168378200531005859375, 332525673007965087890625

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 15), L(1, 15), P(1, 15), T(1, 15). Essentially same as Pisot sequences E(15, 225), L(15, 225), P(15, 225), T(15, 225). See A008776 for definitions of Pisot sequences.
A000005(a(n)) = A000290(n+1). - Reinhard Zumkeller, Mar 04 2007
If X_1, X_2, ..., X_n is a partition of the set {1,2,...,2*n} into blocks of size 2 then, for n>=1, a(n) is equal to the number of functions f : {1,2,..., 2*n}->{1,2,3,4} such that for fixed y_1,y_2,...,y_n in {1,2,3,4} we have f(X_i)<>{y_i}, (i=1,2,...,n). - Milan Janjic, May 24 2007
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 15-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Number of ways to assign truth values to n quaternary disjunctions connected by conjunctions such that the proposition is true. For example, a(2) = 225, since for the proposition (a v b v c v d) & (e v f v g v h) there are 225 assignments that make the proposition true. - Ori Milstein, Jan 26 2023

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 279
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
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 (15).

a(n) = A159991(n)/A000302(n). - Reinhard Zumkeller, May 02 2009

Row 6 of A329332.

[ 15^n: n in [0..20] ]; // Vincenzo Librandi, Nov 21 2010

[ n eq 1 select 1 else 15*Self(n-1): n in [1..21] ];

A001024:=-1/(-1+15*z); # Simon Plouffe in his 1992 dissertation

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

a(n)=15^n \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 1/(1-15x), e.g.f.: exp(15x)

a(n) = 15^n; a(n) = 15*a(n-1) with a(0)=1. - Vincenzo Librandi, Nov 21 2010

More terms from James Sellers, Sep 19 2000

A029858 a(n) = (3^n - 3)/2.

Original entry on oeis.org

0, 3, 12, 39, 120, 363, 1092, 3279, 9840, 29523, 88572, 265719, 797160, 2391483, 7174452, 21523359, 64570080, 193710243, 581130732, 1743392199, 5230176600, 15690529803, 47071589412, 141214768239

Offset: 1

Christian G. Bower

Also the number of 2-block covers of a labeled n-set. a(n) = A055154(n,2). Generally, number of k-block covers of a labeled n-set is T(n,k) = (1/k!)*Sum_{i = 1..k + 1} Stirling1(k + 1,i)*(2^(i - 1) - 1)^n. In particular, T(n,2) = (1/2!)*(3^n - 3), T(n,3) = (1/3!)*(7^n - 6*3^n + 11), T(n,4) = (1/4)!*(15^n - 10*7^n + 35*3^n - 50), ... - Vladeta Jovovic, Jan 19 2001
Conjectured to be the number of integers from 0 to 10^(n-1) - 1 that lack 0, 1, 2, 3, 4, 5 and 6 as a digit. - Alexandre Wajnberg, Apr 25 2005. This is easily verified to be true. - Renzo Benedetti, Sep 25 2008
Number of monic irreducible polynomials of degree 1 in GF(3)[x1,...,xn]. - Max Alekseyev, Jan 23 2006
Also, the greatest number of identical weights among which an odd one can be identified and it can be decided if the odd one is heavier or lighter, using n weighings with a comparing balance. If the odd one only needs to be identified, the sequence starts 4, 13, 40 and is A003462 (3^n - 1)/2, n > 1. - Tanya Khovanova, Dec 11 2006; corrected by Samuel E. Rhoads, Apr 18 2016
Binomial transform yields A134057. Inverse binomial transform yields A062510 with one additional 0 in front. - R. J. Mathar, Jun 18 2008
Numbers k where the recurrence s(0)=0, if s(k-1) >= k then s(k) = s(k-1) - k otherwise s(k) = s(k-1) + k produces s(k) = 0. - Hugo Pfoertner, Jan 05 2012
For n > 1: A008344(a(n)) = a(n). - Reinhard Zumkeller, May 09 2012
Also the number of edges in the (n-1)-Hanoi graph. - Eric W. Weisstein, Jun 18 2017
A level 1 Sierpiński triangle graph 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.  a(n) is the number of degree 4 vertices in the level n Sierpinski triangle graph. - Allan Bickle, Jul 30 2020
Also the number of minimum vertex cuts in the n-Apollonian network. - Eric W. Weisstein, Dec 20 2020
Also the minimum number of turns in n-dimensional Euclidean space needed to visit all 3^n points of the grid {0, 1, 2}^n, moving in straight lines between turns (repeated visits and direction changes at non-grid points are allowed). - Marco Ripà, Aug 06 2025

			For the Sierpiński triangle, Level 1 is a triangle, so a(1) = 0.
Level 2 has three corners (degree 2) and three degree 4 vertices, so a(2) = 3.
The level 2 Hanoi graph has 3 triangles joined by 3 edges, so a(2+1) = 12.

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
Natalia Agudelo Muñetón, Agustín Moreno Cañadas, Pedro Fernando Fernández Espinosa, and Isaías David Marín Gaviria, Brauer Configuration Algebras and Their Applications in Graph Energy Theory, Mathematics (2021) Vol. 9, 3042.
Alex Born, Cor A. J. Hukrnes, and Gerhard J. Woeginger, How to detect a counterfeit coin: adaptive versus non-adaptive solutions, Inf. Proc. Lett. 86 (2003) 137-141.
Gary Darby, The Counterfeit Coin
Madeleine Goertz and Aaron Williams, The Quaternary Gray Code and How It Can Be Used to Solve Ziggurat and Other Ziggu Puzzles, arXiv:2411.19291 [math.CO], 2024. See p. 17.
Lorenz Halbeisen and Norbert Hungerbuhler, The general counterfeit coin problem, Discr. Math 147 (1-3) (1995) 139-150, Theorem 1 with b=1.
Andreas Hinz, Sandi Klavzar, and Sara Sabrina Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
Bennet Manvel, Counterfeit coin problems, Math. Mag. 50 (2) (1977) 90-92, theorem 2.
Marco Ripà, Solving the 106 years old 3^k points problem with the clockwise-algorithm, Journal of Fundamental Mathematics and Applications, 2020, 3(2), 84-97.
Allen Stenger and Jack Wert, The Twelve Coins (or Twelve bags of Gold)
Eric Weisstein's World of Mathematics, Apollonian Network
Eric Weisstein's World of Mathematics, Edge Count
Eric Weisstein's World of Mathematics, Hanoi Graph
Eric Weisstein's World of Mathematics, Minimum Vertex Cut
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A055154, A003462, A007051, A034472, A024023.
Cf. A000203, A008776.
Cf. A007283, A029858, A067771, A233774, A233775, A246959 (Sierpiński triangle graphs).
Cf. A000225, A029858, A058809, A375256 (Hanoi graphs).

a029858 = (`div` 2) . (subtract 3) . (3 ^)
a029858_list = iterate ((+ 3) . (* 3)) 0
-- Reinhard Zumkeller, May 09 2012

[(3^n-3)/2: n in [1..30]]; // Vincenzo Librandi, Jun 05 2011

a:=n->sum(3^j,j=1..n): seq(a(n),n=0..23); # Zerinvary Lajos, Jun 27 2007

Table[(3^n - 3)/2, {n, 24}] (* Alonso del Arte, Dec 29 2014 *)

a(n)=(3^n-3)\2 \\ Charles R Greathouse IV, Apr 17 2012

a(n) = 3*a(n-1) + 3. - Alexandre Wajnberg, Apr 25 2005
O.g.f: 3*x^2/((1-x)*(1-3*x)). - R. J. Mathar, Jun 18 2008
a(n) = 3^(n-1) + a(n-1) (with a(1)=0). - Vincenzo Librandi, Nov 18 2010
a(n) = 3*A003462(n-1). - R. J. Mathar, Sep 10 2015
E.g.f.: 3*(-1 + exp(2*x))*exp(x)/2. - Ilya Gutkovskiy, Apr 19 2016
a(n) = A067771(n-1) - 3. - Allan Bickle, Jul 30 2020
a(n) = sigma(A008776(n-2)) for n>=2. - Flávio V. Fernandes, Apr 20 2021

Corrected by T. D. Noe, Nov 07 2006

cp's OEIS Frontend

A001021 Powers of 12.

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A048473 a(0)=1, a(n) = 3*a(n-1) + 2; a(n) = 2*3^n - 1.

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A057711 a(0)=0, a(1)=1, a(n) = n*2^(n-2) for n >= 2.

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A009975 Powers of 31: a(n) = 31^n.

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A009973 Powers of 29.

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A009964 Powers of 20.

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A048473 a(0)=1, a(n) = 3a(n-1) + 2; a(n) = 23^n - 1.