

A133494


Diagonal of the array of iterated differences of A047848.


68



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
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OFFSET

0,3


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 nth 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


LINKS



FORMULA

G.f.: (1  2*x)/(1  3*x).
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/(1x)) = 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..n1} a(k))  1 = 3^(n1).  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


EXAMPLE

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)


MAPLE

a:= n> ceil(3^(n1)):


MATHEMATICA



PROG

(PARI) Vec((12*x)/(13*x) + O(x^100)) \\ Altug Alkan, Oct 30 2015
(Magma) [n eq 0 select 1 else 3^(n1): n in [0..30]]; // G. C. Greubel, Nov 20 2023
(SageMath) [(3^n + 2*int(n==0))//3 for n in range(31)] # G. C. Greubel, Nov 20 2023


CROSSREFS

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.


KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



