A160551 Number of unordered ways of making change for n dollars using coins of denominations 1, 5, 10, and 25.

1, 242, 1463, 4464, 10045, 19006, 32147, 50268, 74169, 104650, 142511, 188552, 243573, 308374, 383755, 470516, 569457, 681378, 807079, 947360, 1103021, 1274862, 1463683, 1670284, 1895465, 2140026, 2404767, 2690488, 2997989, 3328070, 3681531, 4059172, 4461793

Offset

0,2

Comments

a(n) is the number of distinct quadruplets (p, k, d, q) of nonnegative integers satisfying p + 5k + 10d + 25q = 100n.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = [x^(100*n)] 1/((1-x)*(1-x^5)*(1-x^10)*(1-x^25)).

a(n) = (3 + 53*n + 270*n^2 + 400*n^3) / 3.

From Alois P. Heinz, Oct 08 2022: (Start)

a(n) = A001299(100*n).

G.f.: (60*x^3+501*x^2+238*x+1)/(x-1)^4. (End)

Example

There are four ways to make $0.10: (1) 10 pennies, (2) 5 pennies and 1 nickel, (3) 2 nickels, and (4) 1 dime.

Maple

f := 1/(1-x)/(1-x^5)/(1-x^10)/(1-x^25); a := n -> (convert(series(f, x, 100*n+1), polynom)-convert(series(f, x, 100*n), polynom)) /x^(100*n);
a := n -> (3 + 53*n + 270*n^2 + 400*n^3) / 3;

Prog

(PARI) a(n) = {(3 + 53*n + 270*n^2 + 400*n^3) / 3} \\ Andrew Howroyd, Feb 02 2020

Crossrefs

Cf. A001299.

Sequence in context: A258894 A258887 A234484 * A258886 A354563 A354565

Adjacent sequences: A160548 A160549 A160550 * A160552 A160553 A160554

Keyword

nonn,easy

Author

Lee A. Newberg, May 18 2009, Jun 15 2009

Extensions

Terms a(21) and beyond from Andrew Howroyd, Feb 02 2020

Status

approved