A001299 Number of ways of making change for n cents using coins of 1, 5, 10, 25 cents.
1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 9, 9, 9, 9, 9, 13, 13, 13, 13, 13, 18, 18, 18, 18, 18, 24, 24, 24, 24, 24, 31, 31, 31, 31, 31, 39, 39, 39, 39, 39, 49, 49, 49, 49, 49, 60, 60, 60, 60, 60, 73, 73, 73, 73, 73, 87, 87, 87, 87, 87, 103, 103, 103, 103, 103
Offset: 0
Examples
G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 2*x^9 + 4*x^10 + ...
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..10000
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 175
- Gerhard Kirchner, Derivation of formulas
- Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
- Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
- Index entries for sequences related to making change.
- Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 1, -1).
Programs
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Haskell
a001299 = p [1,5,10,25] where p _ 0 = 1 p [] _ = 0 p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m -- Reinhard Zumkeller, Dec 15 2013
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Mathematica
CoefficientList[ Series[ 1 / ((1 - x)(1 - x^5)(1 - x^10)(1 - x^25)), {x, 0, 65} ], x ] Table[Length[FrobeniusSolve[{1,5,10,25},n]],{n,0,80}] (* Harvey P. Dale, Dec 01 2015 *) a[ n_] := With[ {m = Quotient[n, 5] / 10}, Round[ (4 m + 3) (5 m + 1) (5 m + 2) / 6]]; (* Michael Somos, Feb 23 2017 *) -
PARI
a(n)=floor((n\5+1)*((n\5+2)*(2-n%5)/100+[54,27,-2,-33,-66][n%5+1]/500)+(2-5*(n%5%2))*(-1)^n/40+(2*n^3+123*n^2+2146*n+16290)/15000) \\ Tani Akinari, May 09 2014
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PARI
{a(n) = my(m=n\5 / 10); round((4*m + 3) * (5*m + 1) * (5*m + 2) / 6)}; /* Michael Somos, Feb 23 2017 */
Formula
G.f.: 1/((1-x)*(1-x^5)*(1-x^10)*(1-x^25)).
a(n) = round((100*x^3 + 135*x^2 +53*x)/6) + 1 with x= floor(n/5)/10. See link "Derivation of formulas". - Gerhard Kirchner, Feb 23 2017
Comments