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A069755
Frobenius number of the numerical semigroup generated by 3 consecutive triangular numbers.
10
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
OFFSET
2,1
COMMENTS
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.
LINKS
R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).
Aureliano M. Robles-Pérez, José Carlos Rosales, The Frobenius number for sequences of triangular and tetrahedral numbers, arXiv:1706.04378 [math.NT], 2017.
FORMULA
Conjectures from Colin Barker, Nov 22 2012: (Start)
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*(17 + 12*x + 9*x^2 - 3*x^4 + x^6) / ((1 - x)^4*(1 + x)^3). (End)
Conjectures from Colin Barker, Mar 21 2017: (Start)
a(n) = (6*n^3 + 18*n^2 + 12*n - 8)/8 for n even.
a(n) = (6*n^3 + 12*n^2 - 6*n - 20)/8 for n odd. (End)
EXAMPLE
a(2)=17 because 17 is not a nonnegative linear combination of 3, 6 and 10 but all numbers greater than 17 are.
MATHEMATICA
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]] (* Harvey P. Dale, Oct 04 2011 *)
CROSSREFS
KEYWORD
easy,nice,nonn
AUTHOR
Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 05 2002
EXTENSIONS
Corrected by T. D. Noe, Nov 27 2006
STATUS
approved