A106517 Convolution of Fibonacci(n-1) and 3^n.

1, 3, 10, 31, 95, 288, 869, 2615, 7858, 23595, 70819, 212512, 637625, 1913019, 5739290, 17218247, 51655351, 154967040, 464902717, 1394710735, 4184136386, 12552415923, 37657258715, 112971793856, 338915410225, 1016746277043

Offset: 0

Author

Paul Barry, May 05 2005

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 (4,-2,-3).

Crossrefs

Cf. A000032, A000045, A000244, A094688, A101220.

Diagonal sums of number triangle A106516.

Programs

Magma

I:=[1,3,10]; [n le 3 select I[n] else 4*Self(n-1) -2*Self(n-2) -3*Self(n-3): n in [1..41]]; // G. C. Greubel, Aug 05 2021

Mathematica

LinearRecurrence[{4,-2,-3},{1,3,10},30] (* Harvey P. Dale, Oct 08 2014 *)

PARI

a(n) = sum(k=0, n, fibonacci(n-k-1) * 3^k); \\ Michel Marcus, Aug 06 2021

Sage

[(2*3^(n+1) - lucas_number2(n+1, 1, -1))/5 for n in (0..40)] # G. C. Greubel, Aug 05 2021

Formula

G.f.: (1-x)/((1-x-x^2)*(1-3*x)).

a(n) = Sum_{k=0..n} Fibonacci(n-k-1) * 3^k.

a(n) = A101220(2, 3, n+1). - Ross La Haye, Jul 25 2005

a(n) = A101220(3, 3, n+1) - A101220(3, 3, n). - Ross La Haye, May 31 2006

a(n) = (1/5)*(6*3^n - Lucas(n+1)). - Ralf Stephan, Nov 16 2010

Sum_{k=0..n} a(k) = A094688(n+1). - G. C. Greubel, Aug 05 2021