A069761 Frobenius number of the numerical semigroup generated by four consecutive tetrahedral numbers.

41, 249, 253, 853, 1243, 1571, 2619, 5059, 5357, 9437, 11801, 13609, 18327, 27607, 28919, 41951, 49169, 54473, 67253, 90573, 94051, 124099, 140347, 152027, 178989, 226141, 233369, 291089, 321839, 343639, 392631, 475999, 488993, 587633, 639653, 676181, 756779

Offset: 2

Author

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 09 2002

Keywords

• easy
• nonn
• changed

Comments

The Frobenius number of a 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. Since four consecutive tetrahedral numbers are relatively prime, they generate a numerical semigroup with a Frobenius number.

Examples

			a(2) = 41 because 41 is not a nonnegative linear combination of 4, 10, 20 and 35, but all integers greater than 43 are.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 2..100
• R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).

Crossrefs

Cf. A000292, A037165, A059769, A069755-A069764.

Programs

Mathematica

FrobeniusNumber/@Partition[Binomial[Range[2,50]+2,3],4,1] (* Harvey P. Dale, Jan 22 2012 *)

Formula

Conjecture: a(n)= +a(n-1) +4*a(n-6) -4*a(n-7) -6*a(n-12) +6*a(n-13) +4*a(n-18) -4*a(n-19) -a(n-24) +a(n-25). - R. J. Mathar, Aug 15 2025

Conjectured g.f.: x^2*(-4*x^2 -600*x^3 -390*x^4 -1680*x^9 -282*x^8 -496*x^11 -804*x^10 -208*x -312*x^15 -144*x^14 -768*x^13 -772*x^12-41 -32*x^18 -40*x^17 -102*x^16 -2*x^20 -8*x^19 -1608*x^7 +x^24 -884*x^6 -328*x^5) / ( (1+x)^4 *(x^2-x+1)^4 *(1+x+x^2)^4 *(x-1)^5 ). - R. J. Mathar, Aug 15 2025

Extensions

Sequence terms corrected and extended by Harvey P. Dale, Jan 22 2012

Offset corrected and example corrected by Harvey P. Dale, Jan 24 2012