A360692 a(0) = 0. Thereafter a(n+1) = a(a(n)) if a(n) has not occurred previously, otherwise a(n+1) = n - 1 - a(n-1).

0, 0, 0, 1, 0, 2, 0, 3, 1, 4, 0, 5, 2, 6, 0, 7, 3, 8, 1, 9, 4, 10, 0, 11, 5, 12, 2, 13, 6, 14, 0, 15, 7, 16, 3, 17, 8, 18, 1, 19, 9, 20, 4, 21, 10, 22, 0, 23, 11, 24, 5, 25, 12, 26, 2, 27, 13, 28, 6, 29, 14, 30, 0, 31, 15, 32, 7, 33, 16, 34, 3, 35, 17, 36, 8

Offset: 0

Author

David James Sycamore, Feb 16 2023

Keywords

• nonn

Comments

An inductive argument shows that a(n) <= n for all n, with equality iff n = 0. It follows that a(n) is well defined, and the sequence is infinite.

Apart from a(1) = 0 every repeat term is followed by a novel term, and vice versa.

Every nonnegative integer appears infinitely many times.

The proper subsequence given by a(2*k) for k >= 2 is the sequence itself, which is therefore fractal.

Starting from a(1) = 0 the sequence is the nonnegative integers interleaved with itself.

Examples

			a(0) = 0 is a novel term so a(1) = a(a(0)) = 0. Since a(1) is a repeat term a(2) = 0 - a(0) = 0 - 0 = 0. a(1,2) = 0,0 is the only case of consecutive repeat terms.
Since a(2) = 0 is a repeat term, a(3) = 1 - a(1) = 1 - 0 = 1, a novel term so a(4) = a(a(1)) = 0, and so on.
a(16) = 3, a repeat term (last seen at a(7)), so a(17) = 15 - a(15) = 15 - 7 = 8.
		

Links

• Winston de Greef, Table of n, a(n) for n = 0..10000
• Rémy Sigrist, C program

Crossrefs

Cf. A000225, A027383, A051633, A101279, A176488, A176449.

Programs

C

// See Links section.

Mathematica

nn = 10^4; c[] = False; a[0] = i = j = 0; Reap[Do[If[c[j], k = n - 2 - i, k = a[j]]; Set[{a[n], c[j], i, j}, {k, True, j, k}]; Sow[k], {n, nn}]][[-1, -1]] (* _Michael De Vlieger, Feb 18 2023 *)

Formula

a(2*n + 1) = n for all n >= 0.

A027383(n) = 0. (n >= 0) gives the positions of all zeros after a(0) = 0.

a((2*k + 3)*2^n - 2) = k (n >= 0) gives the positions of all k > 0.

The number of nonnegative terms occurring between consecutive zeros is 0,0,1,1,3,3,7,7,15,15,... (A000225(n), repeat).

a(n) = A101279(n+2) - 1. - Rémy Sigrist, Feb 18 2023

Extensions

More terms from Rémy Sigrist, Feb 18 2023