This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A049835 #29 Jan 07 2025 05:57:19
%S A049835 1,3,7,11,19,21,35,37,49,53,75,65,99,93,105,115,151,127,179,153,181,
%T A049835 193,239,191,257,249,271,261,339,263,375,329,361,373,401,351,487,441,
%U A049835 461,427,563,443,603,517,535,585,683,533,697,619,685,661,811,657,781,711
%N A049835 a(n) = Sum_{k=1..n} T(n,k), array T as in A049834.
%C A049835 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
%H A049835 C. Aistleitner, B. Borda, and M. Hauke, <a href="https://arxiv.org/abs/2210.14095">On the distribution of partial quotients of reduced fractions with fixed denominator</a>, ArXiv preprint arXiv:2210.14095 [math.NT], October 25 2022.
%H A049835 M. Shrader-Frechette, <a href="https://www.jstor.org/stable/2690435">Modified Farey sequences and continued fractions</a>, Math. Mag., 54 (1981), 60-63.
%H A049835 A. C. Yao and D. E. Knuth, <a href="https://doi.org/10.1073/pnas.72.12.4720">Analysis of the subtractive algorithm for greatest common divisors</a>, Proc. Nat. Acad. Sci. USA 72 (1975), 4720-4722.
%F A049835 Yao and Knuth proved that a(n) is asymptotically (6/Pi)^2*n*(log n)^2. - _Jeffrey Shallit_, Jan 31 2023
%p A049835 a:= n-> add(add(i, i=convert(k/n, confrac)), k=1..n):
%p A049835 seq(a(n), n=1..60); # _Alois P. Heinz_, Jan 31 2023
%t A049835 T[n_, k_] := T[n, k] = Which[n < 1 || k < 1, 0, n == k, 1, n < k, T[k, n], True, 1 + T[k, n - k]];
%t A049835 a[n_] := Sum[T[n, k], {k, 1, n}];
%t A049835 Table[a[n], {n, 1, 56}] (* _Jean-François Alcover_, Jan 07 2025 *)
%Y A049835 Cf. A049834, A360264.
%K A049835 nonn
%O A049835 1,2
%A A049835 _Clark Kimberling_