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A105392
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Frobenius number of the subsemigroup of the natural numbers generated by successive pairs of Lucas numbers.
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1
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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
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OFFSET
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1,2
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COMMENTS
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Prime values of this are a(n) where n = 2, 3, 4, 8, 12, 16, 19, 28, 30. Semiprime values of this are a(n) where n = 5, 9, 10, 11, 14, 15, 20, 21, 27, 32.
See the b-file for A000204 for an extended list of Lucas numbers.
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REFERENCES
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R. Froberg, C. Gottlieb and R. Haggkvist, "On numerical semigroups", Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).
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LINKS
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Table of n, a(n) for n=1..26.
Eric Weisstein's World of Mathematics, Lucas numbers.
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FORMULA
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a(n)=(L(n)-1)*(L(n+1)-1)-1 where L(n) is the n-th Lucas number 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]
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EXAMPLE
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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.
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MAPLE
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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
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CROSSREFS
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Cf. A000204, A059769.
Sequence in context: A180502 A261515 A171838 * A090857 A287804 A149657
Adjacent sequences: A105389 A105390 A105391 * A105393 A105394 A105395
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KEYWORD
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easy,nonn
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AUTHOR
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Jonathan Vos Post, May 01 2005
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STATUS
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approved
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