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

0,6

Comments

a(n) = A001300(n) = A169718(n) for n < 50. - Reinhard Zumkeller, Dec 15 2013

Number of partitions of n into parts 1, 5, 10, and 25. - 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..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).

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

Example

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 + ...

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 *)

Prog

(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
(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
(PARI) {a(n) = my(m=n\5 / 10); round((4*m + 3) * (5*m + 1) * (5*m + 2) / 6)}; /* Michael Somos, Feb 23 2017 */

Crossrefs

Cf. A001300, A169718, A000008.

Sequence in context: A105674 A130496 A187243 * A001300 A169718 A001306

Adjacent sequences: A001296 A001297 A001298 * A001300 A001301 A001302

Keyword

nonn,easy

Author

N. J. A. Sloane, Mar 15 1996

Status

approved