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A105392 Frobenius number of the subsemigroup of the natural numbers generated by successive pairs of Lucas numbers.

%I A105392 #29 Aug 15 2025 04:28:30
%S A105392 0,5,17,59,169,475,1287,3449,9149,24155,63557,166919,437839,1147645,
%T A105392 3006777,7875419,20623889,54003395,141397847,370208849,969258949,
%U A105392 2537616955,6643671117,17393524559,45537109919,119218140725
%N A105392 Frobenius number of the subsemigroup of the natural numbers generated by successive pairs of Lucas numbers.
%H A105392 R. Fröberg, C. Gottlieb and R. Häggkvist, <a href="https://gdz.sub.uni-goettingen.de/id/PPN362162808_0035">On numerical semigroups</a>, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).
%H A105392 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasNumber.html">Lucas numbers.</a>
%F A105392 a(n) = (L(n)-1)*(L(n+1)-1)-1 where L(n) = A000204(n).
%F A105392 a(n) = A002878(n)-A000204(n+2)+(-1)^n, for n>1. [_Ralf Stephan_, Nov 15 2010, index shifted by _R. J. Mathar_, Nov 16 2010]
%F A105392 G.f.: x^2*(5+2*x+3*x^2-x^4)/(1+x)/(1-3*x+x^2)/(1-x-x^2). [_Colin Barker_, Feb 17 2012]
%e A105392 a(3) = 17 because the 3rd and 4th Lucas numbers are 4 and 7, so
%e A105392 a(3) = (4-1)*(7-1)-1 = 17. Or, a(3)=17 because 17 is the largest positive
%e A105392 integer that is not a nonnegative linear combination of 4 and 7.
%p A105392 A000204 := proc(n) option remember; if n = 1 then 1; elif n = 2 then 3; else procname(n-1)+procname(n-2) ; end if; end proc:
%p A105392 A105392 := proc(n) A000204(2*n+1)-A000204(n+2)+(-1)^n ; end proc:
%p A105392 seq(A105392(n),n=0..20) ; # _R. J. Mathar_, Nov 16 2010
%Y A105392 Cf. A000204, A059769.
%K A105392 easy,nonn
%O A105392 1,2
%A A105392 _Jonathan Vos Post_, May 01 2005