A214734 Sum_{k=1..n} floor(k*p/q), where (p,q) are either coprime positive integers or q=1 or p=1, n*p>=q, ordered by (n + p + q) ascending, then n ascending, then p ascending.
1, 2, 3, 3, 1, 6, 6, 1, 4, 9, 2, 12, 10, 5, 1, 4, 12, 1, 18, 4, 20, 15, 1, 2, 6, 15, 3, 8, 24, 2, 30, 6, 30, 21, 1, 7, 1, 3, 7, 18, 30, 1, 5, 14, 40, 3, 45, 9, 42, 28, 1, 3, 8, 1, 4, 21, 1, 3, 7, 14, 36, 50, 2, 8, 21, 60, 5, 63, 12, 56, 36
Offset: 1
Keywords
Examples
a(n, 1, 3) = n*(n+1)/ 6 - floor(n/3) - Sum_{k=1..(n mod 3)} (k mod 3) = n*(n+1)/ 6 - floor(n/3) - (4 mod 3)/3 = A130518(n).
Example of the ordering (n,p,q): (1,1,1), (1,1,2), (1,2,1), (2,1,1), (1,1,3), (1,3,1), (2,1,2), (2,2,1), (3,1,1), (1,1,4), ...
Formula
a(n, p, q) = Sum_{k=1..n} floor(k*p/q) defines the sequence.
a(n, p, q) = n*(n+1)*p/q/2 - floor(n/q) * (q-1)/2 - Sum_{k=1...(n mod q)} (k*p mod q)/q (the remaining sum has at most q-1 terms, and can assume at most q values when n varies, i.e., that sum for n is equal to the sum for n+q, so the computation of a(n, p, q) requires adding at most (q+1) terms). [Renzo Benedetti, Jul 27 2012]
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