A132664 a(1)=1, a(2)=2, a(n) = a(n-1) + n 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, 5, 4, 3, 9, 8, 7, 6, 16, 15, 14, 13, 12, 11, 10, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 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, 48

Offset: 1

Author

Hieronymus Fischer, Sep 15 2007

Keywords

• nonn

Comments

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

Also: a(1)=1, a(2)=2, a(n) = a(n-1) - 1, if a(n-1) - 1 > 0 and has not been encountered so far, else a(n) = a(n-1) + n.

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

Crossrefs

Cf. A000032, A102460, A130241.

For an analog concerning Fibonacci numbers see A132665.

See A132666-A132674 for sequences with a similar recurrence rule.

Formula

G.f.: g(x) = (L'(x) - x^2 - 1/(1-x))/(1-x) where L(x) = Sum_{k>=0} x^Lucas(k) and Lucas(k) = A000032(k). L(x) is the g.f. of the Lucas indicator sequence (see A102460) and L'(x) = derivative of L(x).

a(n) = Lucas(Lucas_inverse(n+1)+2) - n - 3 = A000032(A130241(n+1) + 2) - n - 3 for n > 1.

a(n) = A000032(floor(log_phi(n + 3/2)) + 2) - n - 3 for n > 1, where phi = (1 + sqrt(5))/2 is the golden ratio.