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A123918 a(n) = F(L(n)) - L(F(n)) where F(n) = n-th Fibonacci number and L(n) = n-th Lucas number. Commutator [Fibonacci, Lucas] at n.

%I A123918 #12 Sep 08 2022 08:45:28
%S A123918 -1,0,1,0,9,78,2537,513708,2971190597,3416454610154664,
%T A123918 22698374052006551837970693,
%U A123918 173402521172797813159681057129399205126250,8801063578447437644962364569698707633118370038189051093925447758629
%N A123918 a(n) = F(L(n)) - L(F(n)) where F(n) = n-th Fibonacci number and L(n) = n-th Lucas number. Commutator [Fibonacci, Lucas] at n.
%C A123918 a(0) = -1 is the only negative value.
%H A123918 G. C. Greubel, <a href="/A123918/b123918.txt">Table of n, a(n) for n = 0..17</a>
%F A123918 a(n) = A068096(n) - A068098(n).
%F A123918 a(n) = Commutator [A000045, A000032] at n.
%F A123918 a(n) = A000045(A000032(n)) - A000032(A000045(n)).
%e A123918 a(0) = F(L(0)) - L(F(0)) = F(2) - L(0) = 1 - 2 = -1.
%e A123918 a(1) = F(L(1)) - L(F(1)) = F(1) - L(1) = 1 - 1 = 0.
%e A123918 a(2) = F(L(2)) - L(F(2)) = F(3) - L(1) = 2 - 1 = 1.
%e A123918 a(3) = F(L(3)) - L(F(3)) = F(4) - L(2) = 3 - 3 = 0.
%e A123918 a(4) = F(L(4)) - L(F(4)) = F(7) - L(3) = 13 - 4 = 9.
%e A123918 a(5) = F(L(5)) - L(F(5)) = F(11) - L(5) = 89 - 11 = 78.
%e A123918 a(6) = F(L(6)) - L(F(6)) = F(18) - L(8) = 2584 - 47 = 2537.
%e A123918 a(7) = F(L(7)) - L(F(7)) = F(29) - L(13) = 514229 - 521 = 513708.
%e A123918 a(8) = F(L(8)) - L(F(8)) = 2971215073 - 24476 = 2971190597.
%e A123918 a(9) = F(L(9)) - L(F(9)) = 3416454622906707 - 12752043 = 3416454610154664.
%e A123918 a(10) = F(L(10)) - L(F(10)) = 22698374052006863956975682 - 312119004989 = 22698374052006551837970693.
%e A123918 a(11) = F(L(11)) - L(F(11)) = 173402521172797813159685037284371942044301 - 3980154972736918051 = 173402521172797813159681057129399205126250.
%t A123918 Table[Fibonacci[LucasL[n]]-LucasL[Fibonacci[n]],{n,0,15}] (* _Harvey P. Dale_, Mar 27 2019 *)
%o A123918 (PARI) vector(15, n, n--; f=fibonacci; f(f(n-1)+f(n+1)) - f(f(n)-1) - f(f(n)+1)) \\ _G. C. Greubel_, Aug 06 2019
%o A123918 (Magma) [Fibonacci(Lucas(n)) - Lucas(Fibonacci(n)): n in [0..15]]; // _G. C. Greubel_, Aug 06 2019
%o A123918 (Sage) [fibonacci(lucas_number2(n,1,-1)) - lucas_number2(fibonacci(n),1, -1) for n in (0..15)] # _G. C. Greubel_, Aug 06 2019
%o A123918 (GAP) List([0..15], n->  Fibonacci(Lucas(1,-1,n)[2]) - Lucas(1,-1,Fibonacci(n))[2] ); # _G. C. Greubel_, Aug 06 2019
%Y A123918 Cf. A000032, A000045, A068096, A068098.
%K A123918 easy,sign
%O A123918 0,5
%A A123918 _Jonathan Vos Post_, Oct 28 2006
%E A123918 One additional term from _Harvey P. Dale_, Mar 27 2019