A087908 Largest integer not expressible as a nonnegative linear combination of n and n^2 + 1.
-1, 3, 17, 47, 99, 179, 293, 447, 647, 899, 1209, 1583, 2027, 2547, 3149, 3839, 4623, 5507, 6497, 7599, 8819, 10163, 11637, 13247, 14999, 16899, 18953, 21167, 23547, 26099
Offset: 1
Examples
For n=2, we have to consider nonnegative linear combinations of 2 and 5. Now 3 is not such a combination, but 4=2*2 and 5=1*5 and any positive integer greater than 3 is of the form 2a+b where a and b are nonnegative integers with b equal to 4 or 5. Therefore a(2)=3.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..10000
- M. Beck and S. Zack, Refined upper bounds for the Diophantine problem of Frobenius, arXiv:math/0305420 [math.NT], 2003-2005.
- A. Brauer and J.E. Shockley, On a Problem of Frobenius, J. reine angew. Math. 211(1962), 215-220.
- Robert W. Owens, An Algorithm to Solve the Frobenius Problem, Math. Mag. 76(2003), 264-275.
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Crossrefs
Cf. A064808.
Programs
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Magma
[n^3-n^2-1: n in [1..35]]; // Vincenzo Librandi, Jul 20 2011
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Maple
with (combinat):seq(n^3-fibonacci(3, n), n=1..29); # Zerinvary Lajos, May 25 2008
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Mathematica
Table[n^3-n^2-1,{n,100}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *) LinearRecurrence[{4,-6,4,-1},{-1,3,17,47},100] (* Harvey P. Dale, Jul 19 2011 *) -
PARI
a(n)=n^3-n^2-1 \\ Charles R Greathouse IV, Aug 01 2011
Formula
a(n) = n^3 - n^2 - 1. [This follows from the well-known fact that the largest integer not expressible as a nonnegative linear combination of a and b is the number ab-a-b. - Matthias Beck (beck(AT)math.sfsu.edu), Sep 22 2005]
a(1)=-1, a(2)=3, a(3)=17, a(4)=47; for n>4, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Jul 19 2011
G.f.: x*(x*((x-1)*x+7)-1)/(x-1)^4. - Harvey P. Dale, Jul 19 2011