A180671 a(n) = Fibonacci(n+6) - Fibonacci(6).

0, 5, 13, 26, 47, 81, 136, 225, 369, 602, 979, 1589, 2576, 4173, 6757, 10938, 17703, 28649, 46360, 75017, 121385, 196410, 317803, 514221, 832032, 1346261, 2178301, 3524570, 5702879, 9227457, 14930344, 24157809, 39088161, 63245978, 102334147, 165580133

Offset: 0

Author

Johannes W. Meijer, Sep 21 2010

Keywords

• nonn
• easy

Comments

The a(n+1) (terms doubled) are the Kn15 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..280
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Crossrefs

Cf. A000045.

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+6)-8); # G. C. Greubel, Jul 13 2019

Magma

[Fibonacci(n+6)-Fibonacci(6): 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+6)-fibonacci(6) od: seq(a(n),n=0..nmax);

Mathematica

f[n_]:= Fibonacci[n+6] - Fibonacci[6]; Array[f, 40, 0] (* or *)
LinearRecurrence[{2,0,-1}, {0,5,13}, 41] (* or *)
CoefficientList[Series[x(3x+5)/(x^3-2x+1), {x,0,40}], x] (* Robert G. Wilson v, Apr 11 2017 *)

PARI

for(n=1,40,print(fibonacci(n+6)-fibonacci(6))); \\ Anton Mosunov, Mar 02 2017

PARI

concat(0, Vec(x*(5+3*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ Colin Barker, Apr 20 2017

Sage

[fibonacci(n+6)-8 for n in (0..40)] # G. C. Greubel, Jul 13 2019

Formula

a(n) = F(n+6) - F(6) with F = A000045.

a(n) = a(n-1) + a(n-2) + 8 for n>1, a(0)=0, a(1)=5, and where 8 = F(6).

From Colin Barker, Apr 13 2012: (Start)

G.f.: x*(5 + 3*x)/((1 - x)*(1 - x - x^2)).

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

a(n) = (-8 + (2^(-n)*((1-sqrt(5))^n*(-9+4*sqrt(5)) + (1+sqrt(5))^n*(9+4*sqrt(5)))) / sqrt(5)). - Colin Barker, Apr 20 2017