A132666 a(1)=1, a(n) = 2*a(n-1) if the minimal positive integer not yet in the sequence is greater than a(n-1), else a(n) = a(n-1)-1.

1, 2, 4, 3, 6, 5, 10, 9, 8, 7, 14, 13, 12, 11, 22, 21, 20, 19, 18, 17, 16, 15, 30, 29, 28, 27, 26, 25, 24, 23, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 94, 93, 92, 91, 90, 89, 88, 87, 86, 85

Offset: 1

Author

Hieronymus Fischer, Aug 24 2007, Sep 15 2007

Keywords

• nonn
• nice

Comments

Also: a(1)=1, a(n) = maximal positive number < a(n-1) not yet in the sequence, if it exists, else a(n) = 2*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) = 2*a(n-1).

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

Almost certainly a duplicate of A132340. - R. J. Mathar, Jun 12 2008

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

For formulas concerning a general parameter p (with respect to the recurrence rule ... a(n)=p*a(n-1) ...) see A132674.

For p=3 to p=10 see A132667 through A132674.

For a similar recurrence rule concerning Fibonacci (A000045) and Lucas numbers (A000032) see A132664 and A132665.

Cf. A027383.

Programs

Haskell

import Data.List (delete)
a132666 n = a132666_list !! (n-1)
a132666_list = 1 : f 1 [2..] where
   f z xs  = y : f y (delete y xs) where
     y | head xs > z = 2 * z
       | otherwise   = z - 1
-- Reinhard Zumkeller, Sep 17 2001

Mathematica

max = 72; f[x_] := Sum[x^(2^k), {k, 0, Ceiling[ Log[2, max]]}]; g[x_] = (x (1 - 2x)/(1 - x) + 2x^2*f'[x^3] + 3/4*(f'[x] - 2x - 1))/(1 - x); Drop[ CoefficientList[ Series[ g[x], {x, 0, max}], x], 1] (* Jean-François Alcover, Dec 01 2011 *)

Formula

G.f.: g(x) = (x(1-2x)/(1-x) + 2x^2*f'(x^3) + 3/4*(f'(x)-2x-1))/(1-x) where f(x) = Sum_{k>=0} x^(2^k) and f'(z) = derivative of f(x) at x = z.

a(n) = 5*2^(r/2) - 3 - n, if both r and s are even, else a(n) = 7*2^((s-1)/2) - 3 - n, where r = ceiling(2*log_2((n+2)/3)) and s = ceiling(2*log_2((n+2)/2) - 1).

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

a(n) = A027383(m) + A027383(m+1) + 1 - n, where m:=max{ k | A027383(k) < n }.

a(A027383(n) + 1) = A027383(n+1).

a(A027383(n)) = A027383(n-1) + 1 for n > 0.