A069760 Frobenius number of the numerical semigroup generated by consecutive centered square numbers.

47, 287, 959, 2399, 5039, 9407, 16127, 25919, 39599, 58079, 82367, 113567, 152879, 201599, 261119, 332927, 418607, 519839, 638399, 776159, 935087, 1117247, 1324799, 1559999, 1825199, 2122847

Offset

1,1

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 centered squares are relatively prime, they generate a numerical semigroup with a Frobenius number. The Frobenius number of a 2-generator semigroup <a,b> is ab-a-b.

References

R. Fröberg, C. Gottlieb and R. Häggkvist, "On numerical semigroups", Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).

Links

Table of n, a(n) for n=1..26.

Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Formula

a(n) = 4n^4+16n^3+20n^2+8n-1

a(1)=47,a(2)=287,a(3)=959,a(4)=2399,a(5)=5039,a(n)=5a(n-1)-10a(n-2) +10a(n-3)-5a(n-4)+a(n-5). [From Harvey P. Dale, Apr 25 2011]

G.f.: x*(47+52*x-6*x^2+4*x^3-x^4)/(1-x)^5. [Colin Barker, Feb 14 2012]

Example

a(1)=47 because 47 is not a nonnegative linear combination of 5 and 13, but all integers greater than 47 are.

Mathematica

Table[4n^4+16n^3+20n^2+8n-1, {n, 30}] (* or *) LinearRecurrence[ {5, -10, 10, -5, 1}, {47, 287, 959, 2399, 5039}, 30] (* Harvey P. Dale, Apr 25 2011 *)

Crossrefs

Cf. A001844, A037165, A059769, A069755-A069764.

Sequence in context: A142164 A201545 A142774 * A140043 A074774 A107611

Adjacent sequences: A069757 A069758 A069759 * A069761 A069762 A069763

Keyword

easy,nonn

Author

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 09 2002

Status

approved