cp's OEIS Frontend

A093143 Expansion of (1-5*x)/(1-10*x).

%I A093143 #49 Aug 23 2024 08:42:15
%S A093143 1,5,50,500,5000,50000,500000,5000000,50000000,500000000,5000000000,
%T A093143 50000000000,500000000000,5000000000000,50000000000000,
%U A093143 500000000000000,5000000000000000,50000000000000000,500000000000000000,5000000000000000000,50000000000000000000,500000000000000000000
%N A093143 Expansion of (1-5*x)/(1-10*x).
%C A093143 Partial sums are A093142. A convex combination of 10^n and 0^n.
%C A093143 a(n) is the number of compositions of even natural numbers in n parts <= 9 (0 is counted as a part); also the number of ways of placing of an even number of indistinguishable objects into n distinguishable boxes with the condition that at most 9 objects can be placed in each box. - _Adi Dani_, May 17 2011
%C A093143 See an A246057 comment with a reference for the k-family satisfying a so-called curious cubic identity involving A246057(k-1), a(k) and A002277(k). - _Wolfdieter Lang_, Feb 07 2017
%H A093143 Adi Dani, <a href="http://oeis.org/wiki/Adi_Dani_/Restricted_compositions_of_natural_numbers">Restricted compositions of natural numbers</a>
%H A093143 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (10).
%F A093143 a(n) = 5*10^n/10 for n > 0.
%F A093143 a(n) = Sum_{k=0..n} A134309(n,k)*5^k = Sum_{k=0..n} A055372(n,k)*4^k. - _Philippe Deléham_, Feb 04 2012
%F A093143 From _Elmo R. Oliveira_, Aug 21 2024: (Start)
%F A093143 E.g.f.: (exp(10*x) + 1)/2.
%F A093143 a(n) = 10*a(n-1) for n > 1. (End)
%e A093143 From _Adi Dani_, May 17 2011: (Start)
%e A093143 a(2)=50: there are 50 compositions of even numbers into 2 parts <= 9:
%e A093143 (0,0);
%e A093143 (0,2),(2,0),(1,1);
%e A093143 (0,4),(4,0),(1,3),(3,1),(2,2);
%e A093143 (0,6),(6,0),(1,5),(5,1),(2,4),(4,2),(3,3);
%e A093143 (0,8),(8,0),(1,7),(7,1),(2,6),(6,2),(3,5),(5,3),(4,4);
%e A093143 (1,9),(9,1),(2,8),(8,2),(3,7),(7,3),(4,6),(6,4),(5,5);
%e A093143 (3,9),(9,3),(4,8),(8,4),(5,7),(7,5),(6,6);
%e A093143 (5,9),(9,5),(6,8),(8,6),(7,7);
%e A093143 (7,9),(9,7),(8,8);
%e A093143 (9,9).
%e A093143 (End)
%e A093143 Curious cubic identities (see a comment above): 1^3 + 5^3 + 3^3 = 153, 16^3 + 50^3 + 33^3 = 165033, 166^3 + 500^3 + 333^3 = 166500333, ... - _Wolfdieter Lang_, Feb 07 2017
%t A093143 Table[Ceiling[1/2*10^n],{n,0,30}] (* _Adi Dani_, Jun 20 2011 *)
%t A093143 Join[{1},NestList[10#&,5,20]] (* _Harvey P. Dale_, Apr 10 2021 *)
%o A093143 (PARI) Vec((1-5*x)/(1-10*x) + O(x^100)) \\ _Altug Alkan_, Nov 01 2015
%Y A093143 Cf. A002277, A055372, A093142, A134309, A246057.
%K A093143 nonn,easy
%O A093143 0,2
%A A093143 _Paul Barry_, Mar 24 2004
%E A093143 a(19)-a(21) from _Elmo R. Oliveira_, Aug 21 2024