A330806 - OEIS

%I #30 Sep 08 2022 08:46:24

%S 1,1,2,3,5,4,3,5,8,13,21,34,38,55,93,86,74,117,87,100,92,97,189,286,

%T 332,475,807,744,455,641,1096,1737,2833,4570,5118,7403,12521,19924,

%U 22483,32445,28972,35461,64433,99894,114380,72823,95699,168522,123786,151873

%N 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).

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

%C a(n)^(1/n) tends to 1.228... - _Vaclav Kotesovec_, Jan 01 2020

%H Vaclav Kotesovec, <a href="/A330806/a330806.jpg">Plot of a(n)^(1/n) for n = 1..1000000</a>

%e 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.

%t 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 *)

%o (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

%o (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

%Y Cf. A000045, A214551, A309665.

%K nonn

%O 1,3

%A _Ctibor O. Zizka_, Jan 01 2020