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

Author

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 05 2002

Keywords

• easy
• nice
• nonn

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.

Examples

			a(2)=17 because 17 is not a nonnegative linear combination of 3, 6 and 10 but all numbers greater than 17 are.
		

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.

Crossrefs

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

Programs

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 *)

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)

Extensions

Corrected by T. D. Noe, Nov 27 2006