A180673 - OEIS

%I #29 Dec 11 2023 11:44:31

%S 0,13,34,68,123,212,356,589,966,1576,2563,4160,6744,10925,17690,28636,

%T 46347,75004,121372,196397,317790,514208,832019,1346248,2178288,

%U 3524557,5702866,9227444,14930331,24157796,39088148,63245965,102334134

%N a(n) = Fibonacci(n+8) - Fibonacci(8).

%C 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.

%H Vincenzo Librandi, <a href="/A180673/b180673.txt">Table of n, a(n) for n = 0..275</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).

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

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

%F G.f.: x*(13 + 8*x)/((1 - x)*(1 - x - x^2)). - _Ilya Gutkovskiy_, Feb 24 2017

%F a(n) = 13*A000071(n+2) + 8*A000071(n+1). - _Bruno Berselli_, Feb 24 2017

%F From _Colin Barker_, Feb 24 2017: (Start)

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

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

%p 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);

%t Fibonacci[8 +Range[0, 40]] -21 (* _G. C. Greubel_, Jul 13 2019 *)

%o (Magma) [Fibonacci(n+8) - Fibonacci(8): n in [0..40]]; // _Vincenzo Librandi_, Apr 24 2011

%o (PARI) concat(0, Vec(x*(13+8*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ _Colin Barker_, Feb 24 2017

%o (PARI) a(n)=fibonacci(n+8)-21 \\ _Charles R Greathouse IV_, Feb 24 2017

%o (SageMath) [fibonacci(n+8)-21 for n in (0..40)] # _G. C. Greubel_, Jul 13 2019

%o (GAP) List([0..40], n-> Fibonacci(n+8)-21) # _G. C. Greubel_, Jul 13 2019

%Y Cf. A000045, A000071.

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

%K nonn,easy

%O 0,2

%A _Johannes W. Meijer_, Sep 21 2010