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%I #93 Nov 22 2023 01:44:49
%S 1,1,3,9,27,81,243,729,2187,6561,19683,59049,177147,531441,1594323,
%T 4782969,14348907,43046721,129140163,387420489,1162261467,3486784401,
%U 10460353203,31381059609,94143178827,282429536481,847288609443,2541865828329,7625597484987
%N Diagonal of the array of iterated differences of A047848.
%C 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
%C 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
%C a(n) is the reptend length of 1/3^(n+1) in decimal. - _Jianing Song_, Nov 14 2018
%C 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
%H G. C. Greubel, <a href="/A133494/b133494.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (3).
%F Binomial transform of A078008. - _Paul Curtz_, Aug 04 2008
%F From _R. J. Mathar_, Nov 11 2008: (Start)
%F G.f.: (1 - 2*x)/(1 - 3*x).
%F a(n) = A000244(n-1), n > 0. (End)
%F From _Philippe Deléham_, Nov 13 2008: (Start)
%F a(n) = Sum_{k=0..n} A112467(n,k)*2^k.
%F a(n) = Sum_{k=0..n} A071919(n,k)*2^k. (End)
%F 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
%F G.f.: 1/(1 - (Sum_{k>=1} (x/(1 - x))^k)). - _Joerg Arndt_, Sep 30 2012
%F For n > 0, a(n) = 2*(Sum_{k=0..n-1} a(k)) - 1 = 3^(n-1). - _J. Conrad_, Oct 29 2015
%F 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
%F Invert transform of A011782(n) = 2^(n-1). Second invert transform of A000012. - _Gus Wiseman_, Jul 19 2020
%F a(n) = ceiling(3^(n-1)). - _Alois P. Heinz_, Jul 26 2020
%e From _Gus Wiseman_, Jul 15 2020: (Start)
%e The a(0) = 1 through a(3) = 9 ways to choose a composition of each part of a composition:
%e () (1) (2) (3)
%e (1,1) (1,2)
%e (1),(1) (2,1)
%e (1,1,1)
%e (1),(2)
%e (2),(1)
%e (1),(1,1)
%e (1,1),(1)
%e (1),(1),(1)
%e (End)
%p a:= n-> ceil(3^(n-1)):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jul 26 2020
%t CoefficientList[Series[(1 - 2 x)/(1 - 3 x), {x, 0, 50}], x] (* _Vladimir Joseph Stephan Orlovsky_, Jun 21 2011 *)
%t Join[{1}, 3^(Range[0, 30])] (* _G. C. Greubel_, Nov 20 2023 *)
%o (PARI) a(n)=max(1,3^(n-1)) \\ _Charles R Greathouse IV_, Jul 07 2011
%o (PARI) Vec((1-2*x)/(1-3*x) + O(x^100)) \\ _Altug Alkan_, Oct 30 2015
%o (Magma) [n eq 0 select 1 else 3^(n-1): n in [0..30]]; // _G. C. Greubel_, Nov 20 2023
%o (SageMath) [(3^n + 2*int(n==0))//3 for n in range(31)] # _G. C. Greubel_, Nov 20 2023
%Y The strict version is A336139.
%Y Splittings of partitions are A323583.
%Y Multiset partitions of partitions are A001970.
%Y Partitions of each part of a partition are A063834.
%Y Compositions of each part of a partition are A075900.
%Y Strict partitions of each part of a strict partition are A279785.
%Y Compositions of each part of a strict partition are A304961.
%Y Strict compositions of each part of a composition are A307068.
%Y Compositions of each part of a strict composition are A336127.
%Y Cf. A000244, A000302, A001906, A006951, A011782, A022811, A071919.
%Y Cf. A072706, A078008, A112467, A182105, A316245, A336128, A336135.
%Y Cf. A339006, A355382, A355384, A355387.
%K nonn,easy
%O 0,3
%A _Paul Barry_, _Paul Curtz_, Dec 23 2007
%E Definition clarified by _R. J. Mathar_, Nov 11 2008
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