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

1, 5, 50, 500, 5000, 50000, 500000, 5000000, 50000000, 500000000, 5000000000, 50000000000, 500000000000, 5000000000000, 50000000000000, 500000000000000, 5000000000000000, 50000000000000000, 500000000000000000, 5000000000000000000, 50000000000000000000, 500000000000000000000

Offset: 0

Author

Paul Barry, Mar 24 2004

Keywords

• nonn
• easy

Comments

Partial sums are A093142. A convex combination of 10^n and 0^n.

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

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

Examples

			From _Adi Dani_, May 17 2011: (Start)
a(2)=50: there are 50 compositions of even numbers into 2 parts <= 9:
(0,0);
(0,2),(2,0),(1,1);
(0,4),(4,0),(1,3),(3,1),(2,2);
(0,6),(6,0),(1,5),(5,1),(2,4),(4,2),(3,3);
(0,8),(8,0),(1,7),(7,1),(2,6),(6,2),(3,5),(5,3),(4,4);
(1,9),(9,1),(2,8),(8,2),(3,7),(7,3),(4,6),(6,4),(5,5);
(3,9),(9,3),(4,8),(8,4),(5,7),(7,5),(6,6);
(5,9),(9,5),(6,8),(8,6),(7,7);
(7,9),(9,7),(8,8);
(9,9).
(End)
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
		

Links

• Adi Dani, Restricted compositions of natural numbers
• Index entries for linear recurrences with constant coefficients, signature (10).

Crossrefs

Cf. A002277, A055372, A093142, A134309, A246057.

Programs

Mathematica

Table[Ceiling[1/2*10^n],{n,0,30}] (* Adi Dani, Jun 20 2011 *)
Join[{1},NestList[10#&,5,20]] (* Harvey P. Dale, Apr 10 2021 *)

PARI

Vec((1-5*x)/(1-10*x) + O(x^100)) \\ Altug Alkan, Nov 01 2015

Formula

a(n) = 5*10^n/10 for n > 0.

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

From Elmo R. Oliveira, Aug 21 2024: (Start)

E.g.f.: (exp(10*x) + 1)/2.

a(n) = 10*a(n-1) for n > 1. (End)

Extensions

a(19)-a(21) from Elmo R. Oliveira, Aug 21 2024