This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A133494 #99 Aug 26 2025 14:25:13
%S A133494 1,1,3,9,27,81,243,729,2187,6561,19683,59049,177147,531441,1594323,
%T A133494 4782969,14348907,43046721,129140163,387420489,1162261467,3486784401,
%U A133494 10460353203,31381059609,94143178827,282429536481,847288609443,2541865828329,7625597484987,22876792454961,68630377364883
%N A133494 Diagonal of the array of iterated differences of A047848.
%C A133494 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 A133494 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 A133494 a(n) is the reptend length of 1/3^(n+1) in decimal. - _Jianing Song_, Nov 14 2018
%C A133494 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 A133494 G. C. Greubel, <a href="/A133494/b133494.txt">Table of n, a(n) for n = 0..1000</a>
%H A133494 Bishal Deb, <a href="https://arxiv.org/abs/2508.13709">Cyclic sieving phenomena via combinatorics of continued fractions</a>, arXiv:2508.13709 [math.CO], 2025. See p. 42.
%H A133494 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (3).
%F A133494 Binomial transform of A078008. - _Paul Curtz_, Aug 04 2008
%F A133494 From _R. J. Mathar_, Nov 11 2008: (Start)
%F A133494 G.f.: (1 - 2*x)/(1 - 3*x).
%F A133494 a(n) = A000244(n-1), n > 0. (End)
%F A133494 From _Philippe Deléham_, Nov 13 2008: (Start)
%F A133494 a(n) = Sum_{k=0..n} A112467(n,k)*2^k.
%F A133494 a(n) = Sum_{k=0..n} A071919(n,k)*2^k. (End)
%F A133494 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 A133494 G.f.: 1/(1 - (Sum_{k>=1} (x/(1 - x))^k)). - _Joerg Arndt_, Sep 30 2012
%F A133494 For n > 0, a(n) = 2*(Sum_{k=0..n-1} a(k)) - 1 = 3^(n-1). - _J. Conrad_, Oct 29 2015
%F A133494 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 A133494 Invert transform of A011782(n) = 2^(n-1). Second invert transform of A000012. - _Gus Wiseman_, Jul 19 2020
%F A133494 a(n) = ceiling(3^(n-1)). - _Alois P. Heinz_, Jul 26 2020
%F A133494 From _Elmo R. Oliveira_, Mar 31 2025: (Start)
%F A133494 E.g.f.: (2 + exp(3*x))/3.
%F A133494 a(n) = 3*a(n-1) for n > 1. (End)
%e A133494 From _Gus Wiseman_, Jul 15 2020: (Start)
%e A133494 The a(0) = 1 through a(3) = 9 ways to choose a composition of each part of a composition:
%e A133494 () (1) (2) (3)
%e A133494 (1,1) (1,2)
%e A133494 (1),(1) (2,1)
%e A133494 (1,1,1)
%e A133494 (1),(2)
%e A133494 (2),(1)
%e A133494 (1),(1,1)
%e A133494 (1,1),(1)
%e A133494 (1),(1),(1)
%e A133494 (End)
%p A133494 a:= n-> ceil(3^(n-1)):
%p A133494 seq(a(n), n=0..30); # _Alois P. Heinz_, Jul 26 2020
%t A133494 CoefficientList[Series[(1 - 2 x)/(1 - 3 x), {x, 0, 50}], x] (* _Vladimir Joseph Stephan Orlovsky_, Jun 21 2011 *)
%t A133494 Join[{1}, 3^(Range[0, 30])] (* _G. C. Greubel_, Nov 20 2023 *)
%o A133494 (PARI) a(n)=max(1,3^(n-1)) \\ _Charles R Greathouse IV_, Jul 07 2011
%o A133494 (PARI) Vec((1-2*x)/(1-3*x) + O(x^100)) \\ _Altug Alkan_, Oct 30 2015
%o A133494 (Magma) [n eq 0 select 1 else 3^(n-1): n in [0..30]]; // _G. C. Greubel_, Nov 20 2023
%o A133494 (SageMath) [(3^n + 2*int(n==0))//3 for n in range(31)] # _G. C. Greubel_, Nov 20 2023
%Y A133494 The strict version is A336139.
%Y A133494 Splittings of partitions are A323583.
%Y A133494 Multiset partitions of partitions are A001970.
%Y A133494 Partitions of each part of a partition are A063834.
%Y A133494 Compositions of each part of a partition are A075900.
%Y A133494 Strict partitions of each part of a strict partition are A279785.
%Y A133494 Compositions of each part of a strict partition are A304961.
%Y A133494 Strict compositions of each part of a composition are A307068.
%Y A133494 Compositions of each part of a strict composition are A336127.
%Y A133494 Cf. A000012, A000244, A000302, A001906, A006951, A011782, A022811, A071919.
%Y A133494 Cf. A072706, A078008, A112467, A182105, A316245, A336128, A336135.
%Y A133494 Cf. A339006, A355382, A355384, A355387.
%K A133494 nonn,easy,changed
%O A133494 0,3
%A A133494 _Paul Barry_, _Paul Curtz_, Dec 23 2007
%E A133494 Definition clarified by _R. J. Mathar_, Nov 11 2008