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%I #25 Mar 15 2023 12:39:59
%S 17,29,89,125,251,323,539,659,989,1169,1637,1889,2519,2855,3671,4103,
%T 5129,5669,6929,7589,9107,9899,11699,12635,14741,15833,18269,19529,
%U 22319,23759,26927,28559,32129,33965,37961,40013,44459,46739,51659
%N Frobenius number of the numerical semigroup generated by 3 consecutive triangular numbers.
%C 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.
%H Harvey P. Dale, <a href="/A069755/b069755.txt">Table of n, a(n) for n = 2..1000</a>
%H R. Fröberg, C. Gottlieb and R. Häggkvist, <a href="http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN001258656">On numerical semigroups</a>, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).
%H Aureliano M. Robles-Pérez, José Carlos Rosales, <a href="https://arxiv.org/abs/1706.04378">The Frobenius number for sequences of triangular and tetrahedral numbers</a>, arXiv:1706.04378 [math.NT], 2017.
%F Conjectures from _Colin Barker_, Nov 22 2012: (Start)
%F a(n) = (-14 + 6*(-1)^n + (3+9*(-1)^n)*n + 3*(5+(-1)^n)*n^2 + 6*n^3)/8.
%F G.f.: x^2*(17 + 12*x + 9*x^2 - 3*x^4 + x^6) / ((1 - x)^4*(1 + x)^3). (End)
%F Conjectures from _Colin Barker_, Mar 21 2017: (Start)
%F a(n) = (6*n^3 + 18*n^2 + 12*n - 8)/8 for n even.
%F a(n) = (6*n^3 + 12*n^2 - 6*n - 20)/8 for n odd. (End)
%e a(2)=17 because 17 is not a nonnegative linear combination of 3, 6 and 10 but all numbers greater than 17 are.
%t 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 *)
%t Rest[FrobeniusNumber/@Partition[Accumulate[Range[50]],3,1]] (* _Harvey P. Dale_, Oct 04 2011 *)
%Y Cf. A000217, A037165, A059769, A069756-A069762.
%K easy,nice,nonn
%O 2,1
%A Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 05 2002
%E Corrected by _T. D. Noe_, Nov 27 2006
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