A316628 a(1)=1, a(2)=2, a(3)=2, a(4)=3; a(n) = a(n-a(n-1))+a(n-1-a(n-2)-a(n-2-a(n-2))) for n > 4.

1, 2, 2, 3, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 8, 8, 8, 8, 9, 10, 10, 11, 11, 11, 12, 13, 13, 13, 13, 13, 13, 14, 15, 15, 16, 16, 16, 17, 18, 18, 18, 18, 19, 20, 20, 21, 21, 21, 21, 21, 21, 21, 22, 23, 23, 24, 24, 24, 25, 26, 26, 26, 26, 27, 28, 28, 29, 29, 29, 29, 29, 30, 31, 31

Offset: 1

Author

Nathan Fox, Jul 08 2018

Keywords

• nonn

Comments

This sequence increases slowly.

k occurs A035612(k) times.

Each Fibonacci number occurs more times than any number before it.

Links

• Nathan Fox, Table of n, a(n) for n = 1..10000
• Nathan Fox, Trees, Fibonacci Numbers, and Nested Recurrences, Rutgers University Experimental Math Seminar, Mar 07, 2019

Crossrefs

Cf. A000045, A005185, A035612, A046699.

Programs

GAP

a:=[1,2,2,3];; for n in [5..80] do a[n]:=a[n-a[n-1]]+a[n-1-a[n-2]-a[n-2-a[n-2]]]; od; a; # Muniru A Asiru, Jul 09 2018

Magma

I:=[1,2,2,3]; [n le 4 select I[n] else Self(n-Self(n-1))+Self(n-1-Self(n-2)-Self(n-2-Self(n-2))): n in [1..100]]; // Vincenzo Librandi, Jul 09 2018

Maple

A316628:=proc(n) option remember: if n <= 0 then 0: elif n = 1 then 1: elif n = 2 then 2: elif n = 3 then 2: elif n = 4 then 3: else A316628(n-A316628(n-1)) + A316628(n-1-A316628(n-2)-A316628(n-2-A316628(n-2))): fi: end:

Formula

a(n+1)-a(n)=1 or 0.

a(n)/n -> C=(sqrt(5)-1)/(sqrt(5)+1).