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A069755
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Frobenius number of the numerical semigroup generated by 3 consecutive triangular numbers.
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10
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17, 29, 89, 125, 251, 323, 539, 659, 989, 1169, 1637, 1889, 2519, 2855, 3671, 4103, 5129, 5669, 6929, 7589, 9107, 9899, 11699, 12635, 14741, 15833, 18269, 19529, 22319, 23759, 26927, 28559, 32129, 33965, 37961, 40013, 44459, 46739, 51659
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OFFSET
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2,1
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COMMENTS
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The Frobenius number of the 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. Any three successive triangular numbers are relatively prime, so they generate a numerical semigroup with a Frobenius number.
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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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Harvey P. Dale, Table of n, a(n) for n = 2..1000
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FORMULA
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Conjecture: a(n) = (-14+6*(-1)^n+(3+9*(-1)^n)*n+3*(5+(-1)^n)*n^2+6*n^3)/8. G.f.: x^2*(x^6-3*x^4+9*x^2+12*x+17)/((x-1)^4*(x+1)^3). [Colin Barker, Nov 22 2012]
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EXAMPLE
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a(2)=17 because 17 is not a nonnegative linear combination of 3, 6 and 10 but all numbers greater than 17 are.
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MATHEMATICA
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tri=Range[40]Range[2, 41]/2; Table[t=CoefficientList[Series[1/(1-x^tri[[n]])/(1-x^tri[[n+1]])/(1-x^tri[[n+2]]), {x, 0, n(n+1)(n+2)}], x]; Last[Position[t, 0]-1][[1]], {n, 2, 33}] - T. D. Noe, Nov 27 2006
Rest[FrobeniusNumber/@Partition[Accumulate[Range[50]], 3, 1]] (* From Harvey P. Dale, Oct 04 2011 *)
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CROSSREFS
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Cf. A000217, A037165, A059769, A069756-A069762.
Sequence in context: A007349 A154616 A196938 * A076727 A146870 A146744
Adjacent sequences: A069752 A069753 A069754 * A069756 A069757 A069758
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 05 2002
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EXTENSIONS
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Corrected by T. D. Noe, Nov 27 2006
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STATUS
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approved
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