A132665 a(1)=1, a(2)=3, 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, 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

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

Paolo Xausa, Table of n, a(n) for n = 1..10000

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) = A000045(floor(log_phi(sqrt(5)*(n+1) + 1) + 2)) - n - 3, where phi = (1 + sqrt(5))/2 is the golden ratio.

a(n) = A000045(floor(log_phi(sqrt(5)*n + 2*phi) + 2)) - n - 3.

Mathematica

A132665[n_] := Fibonacci[Quotient[Log[1 + Sqrt[5]*(n+1)], ArcCsch[2]] + 2] - n - 3;
Array[A132665, 100] (* Paolo Xausa, Jan 27 2026 *)

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