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

%I A069761 #23 Sep 01 2025 17:18:20
%S A069761 41,249,253,853,1243,1571,2619,5059,5357,9437,11801,13609,18327,27607,
%T A069761 28919,41951,49169,54473,67253,90573,94051,124099,140347,152027,
%U A069761 178989,226141,233369,291089,321839,343639,392631,475999,488993,587633,639653,676181,756779
%N A069761 Frobenius number of the numerical semigroup generated by four consecutive tetrahedral numbers.
%C A069761 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.
%H A069761 Harvey P. Dale, <a href="/A069761/b069761.txt">Table of n, a(n) for n = 2..100</a>
%H A069761 R. Fröberg, C. Gottlieb and R. Häggkvist, <a href="https://gdz.sub.uni-goettingen.de/id/PPN362162808_0035">On numerical semigroups</a>, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).
%F A069761 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
%F A069761 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
%e A069761 a(2) = 41 because 41 is not a nonnegative linear combination of 4, 10, 20 and 35, but all integers greater than 43 are.
%t A069761 FrobeniusNumber/@Partition[Binomial[Range[2,50]+2,3],4,1] (* _Harvey P. Dale_, Jan 22 2012 *)
%Y A069761 Cf. A000292, A037165, A059769, A069755-A069764.
%K A069761 easy,nonn,changed
%O A069761 2,1
%A A069761 Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 09 2002
%E A069761 Sequence terms corrected and extended by _Harvey P. Dale_, Jan 22 2012
%E A069761 Offset corrected and example corrected by _Harvey P. Dale_, Jan 24 2012