A182194 a(1)=2, a(n)=a(n-1)^2 if the minimal natural number > 1 not yet in the sequence is greater than a(n-1), else a(n)=a(n-1)-1.
2, 4, 3, 9, 8, 7, 6, 5, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90, 89, 88, 87, 86, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54
Offset: 1
Keywords
Examples
a(2)=4=a(1)^2, since 3>2=a(1) is the minimal number not yet in the sequence (because of a(1)=2); a(15)=19=a(14)-1, since the minimal number not yet in the sequence (=10) is <=a(14)=20. a(10^4)=b(8)+b(7)-10^4-2=877. a(10^6)=b(10)+b(9)-10^6-2= 103539133.
Links
- Hieronymus Fischer, Table of n, a(n) for n = 1..10200
Formula
a(n)=a(n-1)-1, if a(n-1)-1 > 1 is not in the set {a(k)| 1<=k<=n-1}, else a(n)=a(n-1)^2.
a(a(n)-1)=n+1.
If we define b(1)=2, b(2)=3, b(k)=b(k-2)^2+1, we get the sequence 2, 3, 5, 10, 26, 101, 677, 10202, 458330, 104080805, …. The b(k) are those terms a(n) of the original sequence for which a(n+1)=a(n)^2.
With these b(k) we obtain for k>1:
a(b(k)-2)=b(k-1),
a(b(k)-1)=b(k-1)^2.
a(b(k))=b(k-1)^2 - 1.
a(n)=b(m)+b(m-1)-n-2, where m is the least index such that b(m)>n+1 (valid for n>=1).
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