A049835 a(n) = Sum_{k=1..n} T(n,k), array T as in A049834.

1, 3, 7, 11, 19, 21, 35, 37, 49, 53, 75, 65, 99, 93, 105, 115, 151, 127, 179, 153, 181, 193, 239, 191, 257, 249, 271, 261, 339, 263, 375, 329, 361, 373, 401, 351, 487, 441, 461, 427, 563, 443, 603, 517, 535, 585, 683, 533, 697, 619, 685, 661, 811, 657, 781, 711

Offset

1,2

Comments

Also the sum of all the partial quotients in the continued fraction for all rational k/n, for 1 <= k <= n. - Jeffrey Shallit, Jan 31 2023

Links

Table of n, a(n) for n=1..56.

C. Aistleitner, B. Borda, and M. Hauke, On the distribution of partial quotients of reduced fractions with fixed denominator, ArXiv preprint arXiv:2210.14095 [math.NT], October 25 2022.

M. Shrader-Frechette, Modified Farey sequences and continued fractions, Math. Mag., 54 (1981), 60-63.

A. C. Yao and D. E. Knuth, Analysis of the subtractive algorithm for greatest common divisors, Proc. Nat. Acad. Sci. USA 72 (1975), 4720-4722.

Formula

Yao and Knuth proved that a(n) is asymptotically (6/Pi)^2*n*(log n)^2. - Jeffrey Shallit, Jan 31 2023

Maple

a:= n-> add(add(i, i=convert(k/n, confrac)), k=1..n):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 31 2023

Crossrefs

Cf. A049834, A360264.

Sequence in context: A244570 A049645 A167181 * A117510 A160227 A134442

Adjacent sequences: A049832 A049833 A049834 * A049836 A049837 A049838

Keyword

nonn

Author

Clark Kimberling

Status

approved