OFFSET
1,2
COMMENTS
Also: a(1)=1, a(n) = maximal positive integer < a(n-1) not yet in the sequence, if it exists, 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 permutation of the positive integers. The sequence is self-inverse, in that a(a(n)) = n.
FORMULA
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).
CROSSREFS
KEYWORD
nonn
AUTHOR
Hieronymus Fischer, Aug 24 2007, Sep 15 2007
STATUS
approved