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

Author

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

Keywords

• nonn
• easy

Comments

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

Examples

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

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).

Crossrefs

Cf. A001299.

Programs

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;

PARI

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

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)

Extensions

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