A132668 a(1)=1, a(n) = 4*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, 4, 3, 2, 8, 7, 6, 5, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 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, 53, 52, 51, 50, 49

Offset: 1

Author

Hieronymus Fischer, Aug 24 2007, Sep 15 2007, Sep 23 2007

Keywords

• nonn

Comments

Also: a(1)=1, a(n) = maximal positive integer < a(n-1) not yet in the sequence, if it exists, else a(n) = 4*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) = 4*a(n-1).

A permutation of the positive integers. The sequence is self-inverse, in that a(a(n)) = n.

Crossrefs

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 through A132674.

Cf. A133628.

Formula

a(n) = (11*4^(r/2) - 5)/3 - n, if both r and s are even, else a(n) = (23*4^((s-1)/2) - 5)/3 - n, where r = ceiling(2*log_4((3n+4)/7)) and s = ceiling(2*log_4((3n+4)/8)).

a(n) = (4^floor(1 + (k+1)/2) + 7*4^floor(k/2) - 5)/3 - 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).

G.f.: g(x) = (x(1-2x)/(1-x) + 4x^2*f'(x^(7/3)) + (7/16)*(f'(x^(1/3)) - 4x - 1))/(1-x) where f(x) = Sum_{k>=0} x^(4^k) and f'(z) = derivative of f(x) at x = z.

a(n) = A133628(m) + A133628(m+1) + 1 - n, where m:=max{ k | A133628(k)

a(A133628(n) + 1) = A133628(n+1).

a(A133628(n)) = A133628(n-1) + 1 for n > 0.

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