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
Keywords
Links
- 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.
Programs
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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
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Mathematica
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]]; a[n_] := Sum[T[n, k], {k, 1, n}]; Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Jan 07 2025 *)
Formula
Yao and Knuth proved that a(n) is asymptotically (6/Pi)^2*n*(log n)^2. - Jeffrey Shallit, Jan 31 2023
Comments