A180671 - OEIS

%I #46 Nov 16 2024 11:59:25

%S 0,5,13,26,47,81,136,225,369,602,979,1589,2576,4173,6757,10938,17703,

%T 28649,46360,75017,121385,196410,317803,514221,832032,1346261,2178301,

%U 3524570,5702879,9227457,14930344,24157809,39088161,63245978,102334147,165580133

%N a(n) = Fibonacci(n+6) - Fibonacci(6).

%C 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.

%H Vincenzo Librandi, <a href="/A180671/b180671.txt">Table of n, a(n) for n = 0..280</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+6) - F(6) with F = A000045.

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

%F From _Colin Barker_, Apr 13 2012: (Start)

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

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

%F 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

%p 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);

%t f[n_]:= Fibonacci[n+6] - Fibonacci[6]; Array[f, 40, 0] (* or *)

%t LinearRecurrence[{2,0,-1}, {0,5,13}, 41] (* or *)

%t CoefficientList[Series[x(3x+5)/(x^3-2x+1), {x,0,40}], x] (* _Robert G. Wilson v_, Apr 11 2017 *)

%o (Magma) [Fibonacci(n+6)-Fibonacci(6): n in [0..40]]; // _Vincenzo Librandi_, Apr 24 2011

%o (PARI) for(n=1,40,print(fibonacci(n+6)-fibonacci(6))); \\ _Anton Mosunov_, Mar 02 2017

%o (PARI) concat(0, Vec(x*(5+3*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ _Colin Barker_, Apr 20 2017

%o (Sage) [fibonacci(n+6)-8 for n in (0..40)] # _G. C. Greubel_, Jul 13 2019

%o (GAP) List([0..40], n-> Fibonacci(n+6)-8); # _G. C. Greubel_, Jul 13 2019

%Y Cf. A000045.

%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