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

0, 5, 17, 59, 169, 475, 1287, 3449, 9149, 24155, 63557, 166919, 437839, 1147645, 3006777, 7875419, 20623889, 54003395, 141397847, 370208849, 969258949, 2537616955, 6643671117, 17393524559, 45537109919, 119218140725

Offset: 1

Author

Jonathan Vos Post, May 01 2005

Keywords

• easy
• nonn

Examples

			a(3) = 17 because the 3rd and 4th Lucas numbers are 4 and 7, so
a(3) = (4-1)*(7-1)-1 = 17. Or, a(3)=17 because 17 is the largest positive
integer that is not a nonnegative linear combination of 4 and 7.
		

Links

• R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).
• Eric Weisstein's World of Mathematics, Lucas numbers.

Crossrefs

Cf. A000204, A059769.

Programs

Maple

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:
A105392 := proc(n) A000204(2*n+1)-A000204(n+2)+(-1)^n ; end proc:
seq(A105392(n),n=0..20) ; # R. J. Mathar, Nov 16 2010

Formula

a(n) = (L(n)-1)*(L(n+1)-1)-1 where L(n) = A000204(n).

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]

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]