A093142 Expansion of g.f. (1-5*x)/((1-x)*(1-10*x)).

1, 6, 56, 556, 5556, 55556, 555556, 5555556, 55555556, 555555556, 5555555556, 55555555556, 555555555556, 5555555555556, 55555555555556, 555555555555556, 5555555555555556, 55555555555555556, 555555555555555556, 5555555555555555556, 55555555555555555556, 555555555555555555556

Offset: 0

Author

Paul Barry, Mar 24 2004

Keywords

• easy
• nonn

Comments

Second binomial transform of 5*A001045(3n)/3+(-1)^n.

Partial sums of A093143.

A convex combination of 10^n and 1.

In general the second binomial transform of k*Jacobsthal(3n)/3+(-1)^n is 1, 1+k, 1+11k, 1+111k, ... This is the case for k=5.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (11,-10).

Crossrefs

Cf. A001045, A002275, A062397, A093143.

Programs

Mathematica

CoefficientList[Series[(1-5x)/((1-x)(1-10x)),{x,0,20}],x] (* or *) LinearRecurrence[{11,-10},{1,6},20] (* Harvey P. Dale, Aug 23 2014 *)

PARI

{a(n) = (5*10^n+4)/9} \\ Seiichi Manyama, Sep 14 2019

Formula

a(n) = 5*10^n/9 + 4/9.

a(n) = 10*a(n-1) - 4 with a(0)=1. - Vincenzo Librandi, Aug 02 2010

a(n) = 11*a(n-1) - 10*a(n-2), n > 1. - Harvey P. Dale, Aug 23 2014

From Elmo R. Oliveira, Apr 29 2025: (Start)

E.g.f.: exp(x)*(5*exp(9*x) + 4)/9.

a(n) = (A062397(n) + A002275(n))/2. (End)