OFFSET
1,2
COMMENTS
a(n) is the smallest amount (in pence) that cannot be made with fewer than n coins.
The coins included are those in common circulation in the UK: 1p, 2p, 5p, 10p, 20p, 50p, £1 (100p) and £2 (200p).
LINKS
Matthew Scroggs, Table of n, a(n) for n = 1..5005
Index entries for linear recurrences with constant coefficients, signature (2,-1).
FORMULA
From Georg Fischer, Jul 22 2022: (Start)
a(n) = 200*n - 1222 for n >= 7.
O.g.f.: (1 + x + 3*x^2 + 5*x^3 + 10*x^4 + 30*x^5 + 50*x^6 + 100*x^7)/(x - 1)^2. (End)
EXAMPLE
The smallest value that requires 4 coins is 18p (10p, 5p, 2p and 1p). Therefore a(4)=18.
MATHEMATICA
CoefficientList[Series[(1 + x + 3*x^2 + 5*x^3 + 10*x^4 + 30*x^5 + 50*x^6 + 100*x^7)/(x - 1)^2, {x, 0, 64}], x] (* Georg Fischer, Jul 22 2022 *)
PROG
(Python) #
coins = [1, 2, 5, 10, 20, 50, 100, 200]
need = [0]
while True:
....next = len(need)
....n_need = next
....for coin in coins:
........if coin>next:
............break
........n_need = min(n_need, 1+need[next-coin])
....need.append(n_need)
....if n_need == len(seq):
........print(next)
(PARI) Vec((1 + x + 3*x^2 + 5*x^3 + 10*x^4 + 30*x^5 + 50*x^6 + 100*x^7)/(x - 1)^2 + O(x^50)) \\ Felix Fröhlich, Jul 23 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Matthew Scroggs, May 25 2015
EXTENSIONS
More terms from Felix Fröhlich, Jul 23 2022
STATUS
approved