1,2

Also: a(1)=1, a(n)=maximal positive number <a(n-1) not encountered so far, if existing, else a(n)=10*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)=10*a(n-1).

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

The following formulas are given for a general parameter p>2 considering the recurrence rule above (i.e. a(n)=p*a(n-1)...; p=10 for this sequence).

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

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

a(n)=(p^floor(1+(k+1)/2)+(2p-1)*p^floor(k/2)-p-1)/(p-1)-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).

For parameters p=2 to p=9 see A132666 -132673.

For a similar recurrence rule concerning Fibonacci and Lucas numbers see A132664 and A132665.

Sequence in context: A052036 A143473 A055121 this_sequence A070562 A070641 A063661

Adjacent sequences: A132671 A132672 A132673 this_sequence A132675 A132676 A132677

nonn

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