A335999 -id:A335999 - cp's OEIS frontend

A336008 Complement of A335999.

5, 7, 9, 12, 15, 18, 20, 23, 25, 28, 30, 33, 36, 38, 41, 44, 46, 49, 52, 54, 57, 59, 62, 65, 67, 70, 72, 75, 78, 80, 83, 85, 88, 91, 93, 96, 99, 101, 104, 106, 109, 112, 114, 117, 120, 122, 125, 127, 130, 133, 135, 138, 141, 143, 146, 148, 151, 154, 156, 159

Offset: 1

Clark Kimberling, Jul 16 2020

nonn

In general, let u(1) = 1, and let k be a positive integer. Define u(n) = least positive integer not in {u(1), ..., u(n-1), v(1), ..., v(n-1)} and v(n) = n - 1 + k + least positive integer not in {u(1), ..., u(n-1), u(n), v(1), ..., v(n-1)}. As sets, (u(n)) and (v(n)) are disjoint. If k >= -1, let a(n) = u(n) and b(n) = v(n) for all n >= 1, but if k <= -2, let a(n) = u(n) - k + 1 and b(n) = v(n) - k - 1 for all n >= 1. Then every positive integer is in exactly one of the sequences (a(n)) and (b(n)). The difference sequence of (a(n)) consists of 1's and 2's; the difference sequence of (b(n)) consists of 2's and 3's. See A335999 for a guide to related sequences.

Cf. A335999.

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
{a, b} = {{1}, {}}; k = 3;
Do[AppendTo[b, Length[b] + k + mex[Flatten[{a, b}], Last[a]]];
AppendTo[a, mex[Flatten[{a, b}], Last[a]]], {150}]
a  (* A335999 *)
b  (* A336008 *)
(* Peter J. C. Moses, Jul 13 2020 *)

Corrected by Clark Kimberling, Sep 26 2020

A384773 a(1) = 1, a(2) = 1. For n > 2 if a(n-1) = k is a novel term, a(n) = a(n-1-k). Otherwise if a(n-1) is a repeat term a(n) = number of m; 1 <= m <= n-2 such that a(m) = a(n-1).

Original entry on oeis.org

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

Offset: 1

David James Sycamore, Jun 09 2025

a(n) <= n for all n, with equality for n = 1. Same as A364749 until a(12).
The sequence of indices of terms a(n-1-k) following novel terms k (starting: 2,3,4,6,8,10,11,13,...) appears to be A335999.
Records subsequence is A000027, with records occurring at indices 1, A026278.

			a(1) = a(2) = 1 implies a(3) = 1 since 1 has been repeated once. Then a(4) = 2 because now 1 has been repeated twice. Since 2 is a novel term a(5) = a(4-2) = a(2) = 1. Since 1 has been repeated three times a(6) = 3, another novel term so a(7) = a(6-3) = a(3) = 1.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000027, A026278, A335999, A364749.

nn = 2^16; c[] := 0; a[1] = a[2] = 1; c[1]++; {1, 1}~Join~Do[(If[c[#] == 0, k = a[n - # - 1], k = c[#] ]; c[#]++) &[a[n - 1] ]; a[n] = k, {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger, Jun 10 2025 *)

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A336008 Complement of A335999.

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