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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 (list; graph; refs; listen; history; text; internal format)
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 2-generated semigroup <a,b> has the formula ab-a-b.

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), 63-83 (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)^2-n^2-(n+1)^2 = n^4+2*n^3-n^2-2*n-1.

a(n) = Numerator of ((n + 2)! - (n - 2)!)/n!, n >=2. - Artur Jasinski, Jan 09 2007

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

a(n) = (n-1)*n*(n+1)*(n+2) - 1 = A052762(n+2) - 1. - Jean-Christophe 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^3-n^2-2*n-1, n=2..50); # Robert Israel, Nov 01 2015

MATHEMATICA

Table[(n^2-1)((n+1)^2-1)-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*x-6*x^2+4*x^3-x^4)/(1-x)^5) \\ Altug Alkan, Nov 01 2015

CROSSREFS

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

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

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Last modified November 21 22:32 EST 2017. Contains 295054 sequences.