A069764 Frobenius number of the numerical semigroup generated by consecutive octahedral numbers.

89, 773, 3611, 12179, 33349, 78889, 167383, 326471, 595409, 1027949, 1695539, 2690843, 4131581, 6164689, 8970799, 12769039, 17822153, 24441941, 32995019, 43908899, 57678389, 74872313, 96140551, 122221399, 153949249, 192262589, 238212323, 292970411, 357838829

Offset: 2

Author

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 18 2002

Keywords

• nonn
• easy

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 octahedral 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(2)=89 because 89 is not a nonnegative linear combination of 6 and 19 (the second and third octahedral numbers), but all integers greater than 89 are.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 2..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 (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A005900, A037165, A059769, A069755-A069763.

Programs

Mathematica

FrobeniusNumber/@Partition[Rest[Table[(n(2n^2+1))/3,{n,30}]],2,1] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{89,773,3611,12179,33349,78889,167383},30] (* Harvey P. Dale, Nov 19 2015 *)

Formula

a(n) = ((1/3)*n*(2*n^2+1)-1)*((1/3)*(n+1)*(2*(n+1)^2+1)-1)-1.

G.f.: x^2*(89+150*x+69*x^2+20*x^3-13*x^4+6*x^5-x^6)/(1-x)^7. - Colin Barker, Feb 12 2012

a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+ 21*a(n-5)- 7*a(n-6)+a(n-7). - Harvey P. Dale, Nov 19 2015

Extensions

More terms from Carl Najafi, Sep 10 2011