A132665 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

A132665

a(1)=1, a(2)=3, a(n) = a(n-1) + n if the minimal positive integer not yet in the sequencer is greater than a(n-1), else a(n) = a(n-1)-1.

1, 3, 2, 6, 5, 4, 11, 10, 9, 8, 7, 19, 18, 17, 16, 15, 14, 13, 12, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 87, 86, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Also: a(1)=1, a(2)=3, 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)=3, 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.

LINKS

Table of n, a(n) for n=1..72.

FORMULA

G.f.: g(x) = (F'(x) - x^2 - 1/(1-x))/(1-x) where F(x) = Sum_{k>=0} x^Fibonacci(k). F(x) is the g.f. of the Fibonacci indicator sequence (see A104162) and F'(x) = derivative of F(x).

a(n) = Fibonacci(Fibonacci_inverse(n+1) + 2) - n - 3 = A000045(A130233(n+1) + 2) - n - 3.

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

a(n) = A000032(floor(log_phi(sqrt(5)*n + 2*phi) + 2)) - n - 3.

CROSSREFS

Cf. A000045, A104152, A130233.

For an analog concerning Lucas numbers see A132664.

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

Sequence in context: A293056 A131968 A191740 * A255122 A191445 A277880

Adjacent sequences: A132662 A132663 A132664 * A132666 A132667 A132668

KEYWORD

nonn

AUTHOR

Hieronymus Fischer, Sep 15 2007

STATUS

approved