A020699 Expansion of (1-3*x)/(1-5*x).

1, 2, 10, 50, 250, 1250, 6250, 31250, 156250, 781250, 3906250, 19531250, 97656250, 488281250, 2441406250, 12207031250, 61035156250, 305175781250, 1525878906250, 7629394531250, 38146972656250, 190734863281250, 953674316406250

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• easy

Comments

Partial sums are A034478.

Except for the first two terms 1 and 2, these are the integers that satisfy phi(n) = 2*n/5. - Michel Marcus, Jul 14 2015

For n>=1, period of powers of 4 mod 10^n. See A000302. - Martin Renner, Jun 12 2020

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..250
• D Bevan, D Levin, P Nugent, J Pantone, L Pudwell, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510:08036 [math.CO], 2015-2016.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1037
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1037 (archived version of page)
• M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
• Index entries for linear recurrences with constant coefficients, signature (5).

Crossrefs

Cf. A034478, A001045, A020729, A000302.

Programs

Maple

seq(`if`(n=0,1,2*5^(n-1)), n=0..22); # Nathaniel Johnston, Jun 26 2011

Mathematica

CoefficientList[Series[(1 - 3 x)/(1 - 5 x), {x, 0, 22}], x] (* Michael De Vlieger, Jul 14 2015 *)

PARI

Vec((1-3*x)/(1-5*x) + O(x^30)) \\ Michel Marcus, Jul 14 2015

Formula

a(n) = 2*5^(n-1) for n>0.

E.g.f.: (2*exp(5*x)+3)/5; a(n)=(2*5^n+3*0^n)/5. - Paul Barry, Sep 03 2003

a(n) = sum{k=0..n, C(n-1, k)*(Jac(2n-2k)+Jac(2n-2k-1))}+0^n/2, where Jac(n)=A001045(n). - Paul Barry, Jun 07 2005

a(0)=1, a(1)=2, a(n) = 5*a(n-1) for n>=2. [Vincenzo Librandi, Jan 01 2011]

a(n) = A020729(n-1), n>0. - R. J. Mathar, Sep 16 2016