A132667 a(1)=1, a(n) = 3*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, 3, 2, 6, 5, 4, 12, 11, 10, 9, 8, 7, 21, 20, 19, 18, 17, 16, 15, 14, 13, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 120, 119, 118, 117, 116

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) = 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 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. A087503.

Formula

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((2*n+3)/5)), s = ceiling(2*log_3((2*n+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)

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

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

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