A069756 - OEIS

%I #45 Feb 10 2021 08:10:30

%S 23,119,359,839,1679,3023,5039,7919,11879,17159,24023,32759,43679,

%T 57119,73439,93023,116279,143639,175559,212519,255023,303599,358799,

%U 421199,491399,570023,657719,755159,863039,982079,1113023,1256639,1413719,1585079,1771559

%N Frobenius number of the numerical semigroup generated by consecutive squares.

%C 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 consecutive squares are relatively prime, they generate a numerical semigroup with a Frobenius number. The Frobenius number of a 2-generated semigroup <a,b> has the formula ab-a-b.

%C Given the set {n, n+1, n+2, n+3} and starting at n=0, the sum of all possible products of the terms in all possible subsets = a(n+2).  Example for n=5, 5+6+7+8=26; 5(6+7+8)+6*(7+8)+7*8=277; 5*(6*7+6*8+7*8)+6*7*8=1066; 5*6*7*8=1680 and the sum of these 15 possible subsets is 3023 = a(5+2) = a(7). The sum is a(n+2) = n^4 + 10*n^3 + 35*n^2 + 50*n + 23. - _J. M. Bergot_, Apr 17 2013

%H T. D. Noe, <a href="/A069756/b069756.txt">Table of n, a(n) for n = 2..1000</a>

%H R. Fröberg, C. Gottlieb and R. Häggkvist, <a href="http://dx.doi.org/10.1007/BF02573091">On numerical semigroups</a>, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = n^2*(n+1)^2-n^2-(n+1)^2 = n^4+2*n^3-n^2-2*n-1.

%F a(n) = Numerator of ((n + 2)! - (n - 2)!)/n!, n >=2. - _Artur Jasinski_, Jan 09 2007

%F G.f.: x^2*(23+4*x-6*x^2+4*x^3-x^4)/(1-x)^5. [_Colin Barker_, Feb 14 2012]

%F a(n) = (n-1)*n*(n+1)*(n+2) - 1 = A052762(n+2) - 1. - _Jean-Christophe Hervé_, Nov 01 2015

%e a(2)=23 because 23 is not a nonnegative linear combination of 4 and 9, but all integers greater than 23 are.

%p seq(n^4+2*n^3-n^2-2*n-1, n=2..50); # _Robert Israel_, Nov 01 2015

%t Table[(n^2-1)((n+1)^2-1)-1, {n,2,30}] (* _T. D. Noe_, Nov 27 2006 *)

%t FrobeniusNumber/@Partition[Range[2,40]^2,2,1] (* _Harvey P. Dale_, Jul 25 2012 *)

%o (PARI) x='x+O('x^50); Vec(x^2*(23+4*x-6*x^2+4*x^3-x^4)/(1-x)^5) \\ _Altug Alkan_, Nov 01 2015

%Y Cf. A000290, A037165, A059769, A069755, A069757-A069764.

%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