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A339094
Number of (unordered) ways of making change for n US Dollars using the current US denominations of 1$, 2$, 5$, 10$, 20$, 50$ and 100$ bills.
0
1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16, 19, 22, 25, 28, 31, 34, 41, 44, 51, 54, 61, 68, 75, 82, 89, 96, 109, 116, 129, 136, 149, 162, 175, 188, 201, 214, 236, 249, 271, 284, 306, 328, 350, 372, 394, 416, 451, 473, 508, 530, 565, 600, 635, 670, 705, 740, 793, 828, 881, 916
OFFSET
0,3
COMMENTS
Not the same as A001313. First difference appears at A001313(100) being 4562, whereas a(100) is 4563; obviously one more than A001313(100).
Not the same as A057537.
Number of partitions of n into parts 1, 2, 5, 10, 20, 50 and 100.
FORMULA
G.f.: 1/((1-x)*(1-x^2)*(1-x^5)*(1-x^10)*(1-x^20)*(1-x^50)*(1-x^100)).
EXAMPLE
a(5) is 4 because 1+1+1+1+1 = 2+1+1+1 = 2+2+1 = 5.
MATHEMATICA
f[n_] := Length@ IntegerPartitions[n, All, {1, 2, 5, 10, 20, 50, 100}]; Array[f, 75, 0] (* or *)
CoefficientList[ Series[1/((1 - x) (1 - x^2) (1 - x^5) (1 - x^10) (1 - x^20) (1 - x^50) (1 - x^100)), {x, 0, 75}], x] (* or *)
Table[ Length@ FrobeniusSolve[{1, 2, 5, 10, 20, 50, 100}, n]], {n, 0, 75}] (* much slower *)
PROG
(PARI) coins(v[..])=my(x='x); prod(i=1, #v, 1/(1-x^v[i]))
Vec(coins(1, 2, 5, 10, 20, 50, 100)+O(x^99)) \\ Charles R Greathouse IV, Jan 24 2022
KEYWORD
easy,nonn
AUTHOR
Robert G. Wilson v, Nov 25 2020
STATUS
approved