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- #!/usr/bin/ruby
- # Daniel "Trizen" Șuteu
- # Date: 27 November 2019
- # https://github.com/trizen
- # See OEIS sequence:
- # https://oeis.org/A051885
- # The smallest numbers whose sum of digits is n, are numbers of the form r*10^j-1, with r=1..9 and j >= 0.
- # This solution uses the following formula:
- # Sum_{j=0..n} (r*10^j-1) = (r * 10^(n+1) - r - 9*n - 9)/9
- # By letting r=1..9, we get:
- # R(k) = Sum_{r=1..9} Sum_{j=0..n} (r*10^j-1) = 2*(2^n * 5^(n+2) - 7) - 9*n
- # From R(k), we get S(k) as:
- # S(k) = R(k) - Sum_{j=2+(k mod 9) .. 9} (j*10^n-1)
- # S(k) = R(k) - (10-r) * (10^n * (r+9) - 2)/2
- # Simplifying the formula, we get:
- # S(k) = (((r-1)*r + 10) * 10^n - 2*(r + 9*n + 4))/2
- # where:
- # n = floor(k/9)
- # r = 2+(k mod 9)
- # https://projecteuler.net/problem=684
- # Runtime: 0.176s
- const MOD = 1000000007
- func S(k) {
- var n = floor(k/9)
- var r = (k%9 + 2)
- (((r-1)*r + 10) * 10**n - 2*(r + 9*n + 4))/2
- }
- func modular_S(k) {
- var n = floor(k/9)
- var r = (k%9 + 2)
- (((r-1)*r + 10) * Mod(10,MOD)**n - 2*(r + 9*n + 4)) / Mod(2, MOD)
- }
- assert_eq(S(20), 1074)
- assert_eq(S(49), 1999945)
- assert_eq(modular_S(20), 1074)
- assert_eq(modular_S(49), 1999945)
- say map(2..90, {|k|
- modular_S(fib(k))
- }).reduce {|a,b| a + b }
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