A051109 Expansion of g.f. (1+2*x+5*x^2)/(1-10*x^3).

1, 2, 5, 10, 20, 50, 100, 200, 500, 1000, 2000, 5000, 10000, 20000, 50000, 100000, 200000, 500000, 1000000, 2000000, 5000000, 10000000, 20000000, 50000000, 100000000, 200000000, 500000000, 1000000000, 2000000000, 5000000000, 10000000000, 20000000000, 50000000000

Offset: 0

Author

Robert Lozyniak (11(AT)onna.com)

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,10).

Crossrefs

Cf. A117727.

Programs

Magma

[(1 +(n mod 3)^2)*10^Floor(n/3): n in [0..40]]; // G. C. Greubel, Jul 23 2023

Mathematica

a[n_]:= a[n]= If[n<3, Fibonacci[2n+1], 10*a[n-3]];
Table[a[n], {n,0,40}] (* G. C. Greubel, Jul 23 2023 *)

Python

print( [ ((n % 3) ** 2 + 1) * 10**int(n/3) for n in range(100)] )

SageMath

[(1 +(n%3)^2)*10^(n//3) for n in range(41)] # G. C. Greubel, Jul 23 2023

Formula

a(3*n) = 10^n, a(3*n+1) = 2*10^n, a(3*n+2) = 5*10^n.

a(n) = ( 1 + (n mod 3)^2 )*10^floor(n/3). - Justin L. Brown (jlbrown(AT)neo.tamu.edu), Jun 17 2004

G.f.: (1+2*x+5*x^2)/(1-10*x^3). - Philippe Deléham, Apr 08 2013

a(n) = 10*a(n-3) with n>2, a(0)=1, a(1)=2, a(2)=5. - Philippe Deléham, Apr 08 2013

From Amiram Eldar, Jul 27 2023: (Start)

Sum_{n>=0} 1/a(n) = 17/9.

Sum_{n>=0} (-1)^n/a(n) = 7/11. (End)

Extensions

Second formula corrected by Peter C. Lauterbach, Nov 12 2010

New name using g.f. from Joerg Arndt, Jul 23 2023