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A069764
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Frobenius number of the numerical semigroup generated by consecutive octahedral numbers.
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9
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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
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OFFSET
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2,1
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COMMENTS
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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 <a,b> has the formula ab-a-b.
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REFERENCES
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R. Fröberg, C. Gottlieb and R. Häggkvist, "On numerical semigroups", Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).
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LINKS
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FORMULA
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a(n)=((1/3)n(2n^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(2)=89, a(3)=773, a(4)=3611, a(5)=12179, a(6)=33349, a(7)=78889, a(8)=167383, 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
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 18 2002
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EXTENSIONS
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STATUS
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approved
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