A094704 Convolution of Fibonacci(n) and 10^n.

0, 1, 11, 112, 1123, 11235, 112358, 1123593, 11235951, 112359544, 1123595495, 11235955039, 112359550534, 1123595505573, 11235955056107, 112359550561680, 1123595505617787, 11235955056179467, 112359550561797254

Offset: 0

Author

Paul Barry, May 21 2004

Keywords

• easy
• nonn

Comments

The convolution of Fibonacci(n) and k^n for k > 1 has a(n) = (1/k^2 - k -1))*(2*k^(n+1) - k*Lucas(n) - (k+2)*Fibonacci(n)).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..990
• Index entries for linear recurrences with constant coefficients, signature (11,-9,-10).

Crossrefs

Cf. A000032, A000045, A019523.

Programs

Magma

[(10^(n+1) -Fibonacci(n+3) -8*Fibonacci(n+1))/89: n in [0..30]]; // G. C. Greubel, Feb 09 2023

Mathematica

CoefficientList[Series[x/((1-x-x^2)(1-10x)),{x,0,20}],x] (* or *) LinearRecurrence[ {11,-9,-10},{0,1,11},20] (* Harvey P. Dale, Mar 18 2013 *)

SageMath

def A094704(n): return (10^(n+1) -fibonacci(n+3) -8*fibonacci(n+1))/89
[A094704(n) for n in range(31)] # G. C. Greubel, Feb 09 2023

Formula

G.f. : x/((1-10*x)*(1-x-x^2)).

a(n) = (1/89)*( 10^(n+1) - 5*( ((1 + sqrt(5))/2)^n + ((1 - sqrt(5))/2)^n ) - (6/sqrt(5))*( ((1 + sqrt(5))/2)^n - ((1 - sqrt(5))/2)^n ) ).

a(n) = 10*a(n-1) + Fibonacci(n) for n >= 1. - Mark Dols, Aug 31 2009

a(n) = 11*a(n-1) - 9*a(n-2) - 10*a(n-3), n > 2. - Harvey P. Dale, Mar 18 2013

a(n) = (1/89)*( 10^(n+1) - Fibonacci(n+3) - 8*Fibonacci(n+1) ). - G. C. Greubel, Feb 09 2023