A330806 a(1) = 1; a(2) = 1; for n >= 3, a(n) = a(n-1) / gcd(a(n-1), n-1) + a(n-2) / gcd(a(n-2), n-2).

1, 1, 2, 3, 5, 4, 3, 5, 8, 13, 21, 34, 38, 55, 93, 86, 74, 117, 87, 100, 92, 97, 189, 286, 332, 475, 807, 744, 455, 641, 1096, 1737, 2833, 4570, 5118, 7403, 12521, 19924, 22483, 32445, 28972, 35461, 64433, 99894, 114380, 72823, 95699, 168522, 123786, 151873

Offset

1,3

Comments

2 >= a(n) / a(n-1) > 0. Empirically the average growth rate is 1.32... in comparison to 1.618... (golden ratio).

a(n)^(1/n) tends to 1.228... - Vaclav Kotesovec, Jan 01 2020

Links

Table of n, a(n) for n=1..50.

Vaclav Kotesovec, Plot of a(n)^(1/n) for n = 1..1000000

Example

a(1) = 1; a(2) = 1; a(3) = 1/gcd(1,2) + 1/gcd(1,1) = 2; a(4) = 2/gcd(2,3) + 1/gcd(1,2) = 3 and so on.

Mathematica

a[1] = a[2] = 1; a[n_] := a[n] = a[n - 1] / GCD[a[n - 1], n - 1] + a[n - 2] / GCD[a[n - 2], n - 2]; Array[a, 100] (* Amiram Eldar, Jan 01 2020 *)

Prog

(Magma) a:=[1, 1]; for n in [3..60] do Append(~a, a[n-1]/ Gcd(a[n-1], n-1) + a[n-2] / Gcd(a[n-2], n-2)); end for; a; // Marius A. Burtea, Jan 01 2020
(PARI) seq(n)={my(a=vector(n)); a[1]=a[2]=1; for(n=3, #a, a[n] = a[n-1]/gcd(a[n-1], n-1) + a[n-2]/gcd(a[n-2], n-2)); a} \\ Andrew Howroyd, Jan 01 2020

Crossrefs

Cf. A000045, A214551, A309665.

Sequence in context: A068508 A137403 A082233 * A058981 A117339 A096016

Adjacent sequences: A330803 A330804 A330805 * A330807 A330808 A330809

Keyword

nonn

Author

Ctibor O. Zizka, Jan 01 2020

Status

approved