|
%I #42 Apr 10 2021 20:06:38
%S 1,5,50,500,5000,50000,500000,5000000,50000000,500000000,5000000000,
%T 50000000000,500000000000,5000000000000,50000000000000,
%U 500000000000000,5000000000000000,50000000000000000,500000000000000000
%N Expansion of (1-5*x)/(1-10*x).
%C Partial sums are A093142. A convex combination of 10^n and 0^n.
%C 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 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 Adi Dani, <a href="http://oeis.org/wiki/Adi_Dani_/Restricted_compositions_of_natural_numbers">Restricted compositions of natural numbers</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (10).
%F a(n) = 5*10^n/10 for n>0.
%F 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
%e From _Adi Dani_, May 17 2011: (Start)
%e a(2)=50: there are 50 compositions of even numbers into 2 parts <= 9:
%e (0,0);
%e (0,2),(2,0),(1,1);
%e (0,4),(4,0),(1,3),(3,1),(2,2);
%e (0,6),(6,0),(1,5),(5,1),(2,4),(4,2),(3,3);
%e (0,8),(8,0),(1,7),(7,1),(2,6),(6,2),(3,5),(5,3),(4,4);
%e (1,9),(9,1),(2,8),(8,2),(3,7),(7,3),(4,6),(6,4),(5,5);
%e (3,9),(9,3),(4,8),(8,4),(5,7),(7,5),(6,6);
%e (5,9),(9,5),(6,8),(8,6),(7,7);
%e (7,9),(9,7),(8,8);
%e (9,9).
%e (End)
%e 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 Table[Ceiling[1/2*10^n],{n,0,30}] (* _Adi Dani_, Jun 20 2011 *)
%t Join[{1},NestList[10#&,5,20]] (* _Harvey P. Dale_, Apr 10 2021 *)
%o (PARI) Vec((1-5*x)/(1-10*x) + O(x^100)) \\ _Altug Alkan_, Nov 01 2015
%Y Cf. A002277, A246057.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Mar 24 2004
|
