A069759 Frobenius number of the numerical semigroup generated by consecutive hex numbers.

107, 647, 2159, 5399, 11339, 21167, 36287, 58319, 89099, 130679, 185327, 255527, 343979, 453599, 587519, 749087, 941867, 1169639, 1436399, 1746359, 2103947, 2513807, 2980799, 3509999, 4106699, 4776407

Offset: 1

Author

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 08 2002

Keywords

• easy
• nonn

Comments

The Frobenius number of a numerical semigroup generated by relatively prime integers a_1,...,a_n is the largest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since consecutive hex numbers are relatively prime, they generate a numerical semigroup with a Frobenius number. The Frobenius number of a 2-generated semigroup has the formula ab-a-b.

Examples

			a(1)=107 because 107 is not a nonnegative linear combination of 7 and 19, but all integers greater than 107 are.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A003215, A037165, A059769, A069755-A069764.

Programs

Mathematica

FrobeniusNumber/@Partition[Table[3n^2+3n+1,{n,30}],2,1] (* Harvey P. Dale, Dec 25 2018 *)

Formula

a(n) = 9*n^4+36*n^3+45*n^2+18*n-1; with offset 2, a(n) = 9*n^4-9*n^2-1.

G.f.: x*(107+112*x-6*x^2+4*x^3-x^4)/(1-x)^5. - Colin Barker, Feb 14 2012