a(1)=1, a(n) = 3*a(n-1) if the minimal positive integer not encountered so far is greater than a(n-1), else a(n) = a(n-1) - 1.

Also: a(1)=1, a(n) = maximal positive integer < a(n-1) not encountered so far, if it exists, else a(n) = 3*a(n-1).

proposed

editing

a(1)=1, a(n)=3*a(n-1) if the minimal natural number not encountered so far is greater than a(n-1), else a(n)=a(n-1)-1.

Also: a(1)=1, a(n)=maximal positive number <a(n-1) not encountered so far, if existing, else a(n)=3*a(n-1).

Also: a(1)=1, a(n)=a(n-1)-1, if a(n-1)-1>0 and has not been encountered so far, else a(n)=3*a(n-1).

A reordering of the natural numbers. The sequence is self-inverse, in that a(a(n))=n.

G.f.: g(x)=(x(1-2x)/(1-x)+3x^2*f'(x^(5/2))+(5/9)*(f'(x^(1/2))-3x-1))/(1-x) where f(x)=sum{k>=0, x^(3^k)} and f'(z)=derivative of f(x) at x=z.

a(n)=4*3^(r/2)-2-n if both, r and s are even, else a(n)=7*3^((s-1)/2)-2-n, where r=ceiling(2*log_3((2n+3)/5)), s=ceiling(2*log_3((2n+3)/3)-1).

a(n)=(3^floor(1+(k+1)/2)+(5*3^floor(k/2)-4)/2-n, where k=r if r is odd, else k=s (with respect to r and s above; formally, k=((r+s)-(r-s)*(-1)^r)/2).

a(n)=A087503(m)+A087503(m+1)+1-n, where m:=max{ k | A087503(k)<n }.

a(A087503(n)+1)=A087503(n+1).

a(A087503(n))=A087503(n-1)+1 for n>0.

For formulas concerning a general parameter p (with respect to the recurrence rule ... a(n)=p*a(n-1) ...) see A132374. For p=2 to p=10 see A132666 -132674.

approved

For formulae concerning a general parameter p (with respect to the recurrence rule ... a(n)=p*a(n-1) ...) see A132374. For p=2 to p=10 see A132666 -132674.

Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 24 2007, Sep 15 2007, Sep 23 2007

a(n)=A133627(m)+A133627(m+1)+1-n, where m:=max{ k | A133627(k)<n }.

a(A133627(n)+1)=A133627(n+1).

a(A133627(n))=A133627(n-1)+1 for n>0.

Cf. A133627.

nonn,new