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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 (list; graph; refs; listen; history; text; internal format)
 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 Harvey P. Dale, Table of n, a(n) for n = 2..1000 R. Froberg, C. Gottlieb and R. Haggkvist, 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=RangeRange[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][], {n, 2, 33}] - T. D. Noe, Nov 27 2006 Rest[FrobeniusNumber/@Partition[Accumulate[Range], 3, 1]] (* Harvey P. Dale, Oct 04 2011 *) CROSSREFS Cf. A000217, A037165, A059769, A069756-A069762. Sequence in context: A154616 A196938 A225943 * A076727 A146870 A146744 Adjacent sequences:  A069752 A069753 A069754 * A069756 A069757 A069758 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

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Last modified September 21 04:59 EDT 2019. Contains 327253 sequences. (Running on oeis4.)