A093140 Expansion of (1-6*x)/((1-x)*(1-10*x)).

1, 5, 45, 445, 4445, 44445, 444445, 4444445, 44444445, 444444445, 4444444445, 44444444445, 444444444445, 4444444444445, 44444444444445, 444444444444445, 4444444444444445, 44444444444444445, 444444444444444445, 4444444444444444445, 44444444444444444445, 444444444444444444445

Offset: 0

Author

Paul Barry, Mar 24 2004

Keywords

• easy
• nonn

Comments

Second binomial transform of 4*A001045(3n)/3+(-1)^n. Partial sums of A093141. 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=4.

Links

• Index entries for linear recurrences with constant coefficients, signature (11,-10).

Crossrefs

Cf. A001045, A002477, A093141, A102807.

Programs

Mathematica

CoefficientList[Series[(1-6x)/((1-x)(1-10x)),{x,0,30}],x] (* or *) LinearRecurrence[{11,-10},{1,5},30] (* or *) Join[{1},Table[FromDigits[PadLeft[{5},n,4]],{n,30}]] (* Harvey P. Dale, Dec 17 2022 *)

Formula

G.f.: (1-6*x)/((1-x)*(1-10*x)).

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

a(n+1) = (A102807(n+1)-A002477(n))/((Sum_{i=1..n} 2*10^i) + 3). [Roger L. Bagula, May 22 2010]

a(n) = 10*a(n-1)-5 with a(0)=1. - Vincenzo Librandi, Aug 02 2010

a(n) = 11*a(n-1)-10*a(n-2). - Wesley Ivan Hurt, May 20 2021

E.g.f.: exp(x)*(4*exp(9*x) + 5)/9. - Elmo R. Oliveira, Aug 17 2024

Extensions

a(19)-a(22) from Elmo R. Oliveira, Aug 17 2024