

A069756


Frobenius number of the numerical semigroup generated by consecutive squares.


4



23, 119, 359, 839, 1679, 3023, 5039, 7919, 11879, 17159, 24023, 32759, 43679, 57119, 73439, 93023, 116279, 143639, 175559, 212519, 255023, 303599, 358799, 421199, 491399, 570023, 657719, 755159, 863039, 982079, 1113023, 1256639, 1413719, 1585079, 1771559
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OFFSET

2,1


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


LINKS

T. D. Noe, Table of n, a(n) for n = 2..1000
R. Fröberg, C. Gottlieb and R. Haggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 6383 (for definition of Frobenius number).
Index entries for linear recurrences with constant coefficients, signature (5,10,10,5,1).


FORMULA

a(n) = n^2*(n+1)^2n^2(n+1)^2 = n^4+2*n^3n^22*n1.
a(n) = Numerator of ((n + 2)!  (n  2)!)/n!, n >=2.  Artur Jasinski, Jan 09 2007
G.f.: x^2*(23+4*x6*x^2+4*x^3x^4)/(1x)^5. [Colin Barker, Feb 14 2012]
a(n) = (n1)*n*(n+1)*(n+2)  1 = A052762(n+2)  1.  JeanChristophe Hervé, Nov 01 2015


EXAMPLE

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


MAPLE

seq(n^4+2*n^3n^22*n1, n=2..50); # Robert Israel, Nov 01 2015


MATHEMATICA

Table[(n^21)((n+1)^21)1, {n, 2, 30}] (* T. D. Noe, Nov 27 2006 *)
FrobeniusNumber/@Partition[Range[2, 40]^2, 2, 1] (* Harvey P. Dale, Jul 25 2012 *)


PROG

(PARI) x='x+O('x^50); Vec(x^2*(23+4*x6*x^2+4*x^3x^4)/(1x)^5) \\ Altug Alkan, Nov 01 2015


CROSSREFS

Cf. A000290, A037165, A059769, A069755, A069757A069764.
Sequence in context: A042028 A265982 A253679 * A293565 A099068 A220408
Adjacent sequences: A069753 A069754 A069755 * A069757 A069758 A069759


KEYWORD

easy,nice,nonn


AUTHOR

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


EXTENSIONS

Corrected by T. D. Noe, Nov 27 2006


STATUS

approved



