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Partial sums of F(1) - L(1) + F(2) - L(2) + F(3) - L(3) + ..., where F = A000045 and L = A000032.
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%I #13 Mar 22 2024 17:43:21

%S 1,0,1,-2,0,-4,-1,-8,-3,-14,-6,-24,-11,-40,-19,-66,-32,-108,-53,-176,

%T -87,-286,-142,-464,-231,-752,-375,-1218,-608,-1972,-985,-3192,-1595,

%U -5166,-2582,-8360,-4179,-13528,-6763,-21890,-10944,-35420,-17709,-57312,-28655

%N Partial sums of F(1) - L(1) + F(2) - L(2) + F(3) - L(3) + ..., where F = A000045 and L = A000032.

%C The closely related partial sums of L(1) - F(1) + L(2) - F(2) + L(3) - F(3) + .... are given by A355019.

%H G. C. Greubel, <a href="/A355018/b355018.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,-1).

%F a(n) = 2 - 2*F((n+3)/2) if n is odd, a(n) = 2 - F((n+2)/2) if n is even, where F = A000045 (Fibonacci numbers).

%F a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) for n >= 5.

%F G.f.: (1 - x - 2*x^3)/((1 - x)*(1 - x^2 - x^4)).

%F From _G. C. Greubel_, Mar 17 2024: (Start)

%F a(n) = (1/2)*Sum_{j=0..n} ( (1+(-1)^j)*Fibonacci(floor((j+3)/2)) - (1 - (-1)^j)*Lucas(floor((j+1)/2)) ).

%F a(n) = 2 - (1/2)*( (1+(-1)^n)*Fibonacci(floor((n+2)/2)) + 2*(1-(-1)^n)* Fibonacci(floor((n+3)/2)) ). (End)

%e a(0) = 1

%e a(1) = 1 - 1 = 0

%e a(2) = 1 - 1 + 1 = 1

%e a(3) = 1 - 1 + 1 - 3 = -2.

%t f[n_] := Fibonacci[n]; g[n_] := LucasL[n];

%t f1[n_] := If[OddQ[n], 2 - 2 f[(n + 3)/2], 2 - f[(n + 2)/2]]

%t f2 = Table[f1[n], {n, 0, 20}] (* this sequence *)

%t g1[n_] := If[OddQ[n], -2 + 2 f[(n + 3)/2], -2 + f[(n + 8)/2]]

%t g2 = Table[g1[n], {n, 0, 20}] (* A355019 *)

%t LinearRecurrence[{1,1,-1,1,-1}, {1,0,1,-2,0}, 61] (* _G. C. Greubel_, Mar 17 2024 *)

%o (Magma) F:=Fibonacci; [2 - (((n+1) mod 2)*F(Floor((n+2)/2)) + 2*(n mod 2)*F(Floor((n+3)/2))) : n in [0..60]]; // _G. C. Greubel_, Mar 17 2024

%o (SageMath) f=fibonacci; [2 - (((n+1)%2)*f(((n+2)//2)) +2*(n%2)*f((n+3)//2)) for n in range(61)] # _G. C. Greubel_, Mar 17 2024

%Y Cf. A000032, A000045, A355019, A355020, A355021.

%K sign,easy

%O 0,4

%A _Clark Kimberling_, Jun 16 2022