A069755 Frobenius number of the numerical semigroup generated by 3 consecutive triangular numbers.

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

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).

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

Cf. A000217, A037165, A059769, A069756-A069762.

Sequence in context: A154616 A196938 A225943 * A076727 A347530 A146870

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