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A059769 Frobenius number of the subsemigroup of the natural numbers generated by successive pairs of Fibonacci numbers.

%I A059769 #39 Feb 07 2025 18:08:10
%S A059769 1,7,27,83,239,659,1781,4751,12583,33175,87231,228983,600473,1573655,
%T A059769 4122467,10796939,28273519,74031979,193835949,507497759,1328692751,
%U A059769 3478637807,9107313407,23843452463,62423286769,163426800679,427857750891,1120147480451
%N A059769 Frobenius number of the subsemigroup of the natural numbers generated by successive pairs of Fibonacci numbers.
%H A059769 Harvey P. Dale, <a href="/A059769/b059769.txt">Table of n, a(n) for n = 3..1000</a>
%H A059769 R. Fröberg, C. Gottlieb and R. Häggkvist, <a href="http://dx.doi.org/10.1007/BF02573091">On numerical semigroups</a>, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).
%F A059769 a(n) = (F(n)-1)*(F(n+1)-1)-1 where F(n) is the n-th Fibonacci number.
%F A059769 G.f.: x^3*(1+4*x+5*x^2-x^4)/((1+x)*(1-3*x+x^2)*(1-x-x^2)). [_Colin Barker_, Feb 17 2012]
%F A059769 a(n) = F(n)*F(n+1) - F(n+2). - _Clark Kimberling_, Mar 05 2016
%e A059769 a(3)=1 because the 3rd and 4th Fibonacci numbers are 2 and 3, so a(3)=(2-1)(3-1)-1=1. Or, a(3)=1 because 1 is the largest positive integer that is not a nonnegative linear combination of 2 and 3.
%t A059769 Table[(Fibonacci[n]-1)(Fibonacci[n+1]-1)-1, {n,3,28}] (* _T. D. Noe_, Nov 27 2006 *)
%t A059769 f[n_]:=Fibonacci[n]; Table[f[n+1]f[n+2]-f[n+3], {n,2,40}] (* _Clark Kimberling_, Mar 05 2016 *)
%t A059769 FrobeniusNumber/@Partition[Fibonacci[Range[3,40]],2,1] (* _Harvey P. Dale_, Feb 07 2025 *)
%o A059769 (PARI) x='x+O('x^100); Vec(x^3*(1+4*x+5*x^2-x^4)/(1+x)/(1-3*x+x^2)/(1-x-x^2)) \\ _Altug Alkan_, Mar 05 2016
%o A059769 (Magma) [Fibonacci(n+1)*Fibonacci(n+2)-Fibonacci(n+3): n in [2..30]]; // _Vincenzo Librandi_, Mar 06 2016
%Y A059769 Cf. A000045.
%K A059769 nonn,easy
%O A059769 3,2
%A A059769 Victoria A Sapko (vsapko(AT)math.unl.edu), Feb 21 2001
%E A059769 Corrected by _T. D. Noe_, Nov 27 2006