A180673 a(n) = Fibonacci(n+8) - Fibonacci(8).

0, 13, 34, 68, 123, 212, 356, 589, 966, 1576, 2563, 4160, 6744, 10925, 17690, 28636, 46347, 75004, 121372, 196397, 317790, 514208, 832019, 1346248, 2178288, 3524557, 5702866, 9227444, 14930331, 24157796, 39088148, 63245965, 102334134

Offset: 0

Author

Johannes W. Meijer, Sep 21 2010

Keywords

• nonn
• easy

Comments

The a(n+1) (terms doubled) are the Kn17 sums of the Fibonacci(n) triangle A104763. See A180662 for information about these knight and other chess sums.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..275
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Crossrefs

Cf. A000045, A000071.

Cf. A131524 (Kn11), A001911 (Kn12), A006327 (Kn13), A167616 (Kn14), A180671 (Kn15), A180672 (Kn16), A180673 (Kn17), A180674 (Kn18).

Programs

GAP

List([0..40], n-> Fibonacci(n+8)-21); # G. C. Greubel, Jul 13 2019

Magma

[Fibonacci(n+8) - Fibonacci(8): n in [0..40]]; // Vincenzo Librandi, Apr 24 2011

Maple

nmax:=40: with(combinat): for n from 0 to nmax do a(n):=fibonacci(n+8)-fibonacci(8) od: seq(a(n),n=0..nmax);

Mathematica

Fibonacci[8 +Range[0, 40]] -21 (* G. C. Greubel, Jul 13 2019 *)

PARI

concat(0, Vec(x*(13+8*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ Colin Barker, Feb 24 2017

PARI

a(n)=fibonacci(n+8)-21 \\ Charles R Greathouse IV, Feb 24 2017

SageMath

[fibonacci(n+8)-21 for n in (0..40)] # G. C. Greubel, Jul 13 2019

Formula

a(n) = F(n+8) - F(8) with F(n) the Fibonacci numbers A000045.

a(n) = a(n-1) + a(n-2) + 21 for n>1, a(0)=0, a(1)=13, and where 21 = F(8).

G.f.: x*(13 + 8*x)/((1 - x)*(1 - x - x^2)). - Ilya Gutkovskiy, Feb 24 2017

a(n) = 13*A000071(n+2) + 8*A000071(n+1). - Bruno Berselli, Feb 24 2017

From Colin Barker, Feb 24 2017: (Start)

a(n) = (-21 + (2^(-1-n)*((1-sqrt(5))^n*(-47+21*sqrt(5)) + (1+sqrt(5))^n*(47+21*sqrt(5)))) / sqrt(5)).

a(n) = 2*a(n-1) - a(n-3) for n>2. (End)