A133494 Diagonal of the array of iterated differences of A047848.

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

Author

Paul Barry, Paul Curtz, Dec 23 2007

Keywords

• nonn
• easy
• changed

Comments

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

Examples

			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)
		

Links

• 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).

Crossrefs

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.

Programs

Magma

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

Maple

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

Mathematica

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 *)

PARI

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

PARI

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

SageMath

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

Formula

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)

Extensions

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