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

1,2

References

R. Fröberg, C. Gottlieb and R. Häggkvist, "On numerical semigroups", Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).

Links

Table of n, a(n) for n=1..26.

Eric Weisstein's World of Mathematics, Lucas numbers.

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]

Example

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.

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

Crossrefs

Cf. A000204, A059769.

Sequence in context: A180502 A261515 A171838 * A090857 A287804 A149657

Adjacent sequences: A105389 A105390 A105391 * A105393 A105394 A105395

Keyword

easy,nonn

Author

Jonathan Vos Post, May 01 2005

Status

approved