A001343 Number of (unordered) ways of making change for n cents using coins of 5, 10, 20, 50, 100 cents.

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 6, 0, 0, 0, 0, 6, 0, 0, 0, 0, 9, 0, 0, 0, 0, 9, 0, 0, 0, 0, 13, 0, 0, 0, 0, 13, 0, 0, 0, 0, 18, 0, 0, 0, 0, 18, 0, 0, 0, 0, 24, 0, 0, 0, 0, 24, 0, 0, 0, 0, 31, 0, 0, 0, 0

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn

Comments

Number of partitions of n into parts 5, 10, 20, 50, and 100. - Joerg Arndt, Sep 05 2014

References

• R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 316.
• G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 185
• Index entries for linear recurrences with constant coefficients, order 185.
• Index entries for sequences related to making change.

Programs

Maple

1/(1-x^5)/(1-x^10)/(1-x^20)/(1-x^50)/(1-x^100): seq(coeff(series(%,x,n+1),x,n), n=0..80);

Mathematica

nn = 100; CoefficientList[Series[1/((1 - x^5) (1 - x^10) (1 - x^20) (1 - x^50) (1 - x^100)), {x, 0, nn}], x] (* T. D. Noe, Jun 28 2012 *)
Table[Length[FrobeniusSolve[{5,10,20,50,100},n]],{n,0,80}] (* very slow, Harvey P. Dale, May 19 2012 *)

PARI

a(n)=(n%5<1)*floor(((n\10)^4+38*(n\10)^3+476*(n\10)^2+2185*(n\10)+3734)/2400+(n\10+1)*(-1)^(n\10)/160+(n\10\5+1)*[0,0,1,0,-1][n\10%5+1]/10) \\ Tani Akinari, May 14 2014

Formula

G.f.: 1/((1-x^5)*(1-x^10)*(1-x^20)*(1-x^50)*(1-x^100)).