A163874 a(n) = n-a(a(n-3)) with a(0) = a(1) = a(2) = 0.

0, 0, 0, 3, 4, 5, 3, 3, 3, 6, 7, 8, 9, 10, 11, 9, 9, 9, 12, 13, 14, 12, 12, 12, 15, 16, 17, 18, 19, 20, 18, 18, 18, 21, 22, 23, 24, 25, 26, 24, 24, 24, 27, 28, 29, 27, 27, 27, 30, 31, 32, 33, 34, 35, 33, 33, 33, 36, 37, 38, 36, 36, 36, 39, 40, 41, 42, 43, 44, 42, 42, 42, 45, 46, 47

Offset: 0

Author

Daniel Platt (d.platt(AT)web.de), Aug 08 2009, Sep 14 2009

Keywords

• nonn

Comments

A very near generalization of the Hofstadter G-sequence A005206 since it is part of the following family of sequences (which would give for k=1 the original G-sequence):

a(n)=n-a(a(n-k)) with a(0)=a(1)=...=a(k-1)=0 with k=1,2,3... (here k=3) - for general information about that family see A163873) Every a(n) occurs either exactly one or exactly four times (except from the initial values). A block of four occurrences of the same number n is after the first one interrupted by the following two elements: n+1, n+2 (e.g. see from a(18) to a(23): 12, 13, 14, 12, 12, 12).

Since every natural number occurs in the sequence at least once and 0<=a(n)<=n for all n the elements can be ordered in such a way that every n is connected to its a(n) in a tree structure so that:

..a..

..|..

.a(n)

This will give for the first 41 elements the following (quadrary) tree:

.......3..._..................................

...../.|.\..\...............................

..../..|..\....\............................

.../...|...\......\.........................

../....|....\........\........................

.......6.....7........8.......................

.......|.....|........|.......................

.......9.....10.......11......................

....../.\\\../......../.......................

...../...\\\/______/_________...............

..../.....\/______/________..\..............

.../....../\_____/____.....\..\.............

...|......|......./.....\.....\..\............

..12......13....14.....15.....16..17..........

...|\\\.../...../.......\......\...\..........

...|.\\\/____/___......\......\...\.........

...|..\\/___/__..\......\......\...\........

...|...X_____/_..\..\......\......\...\.......

...|../...../..\..\..\......|......|...|......

..18..19..20...21.22.23.....24.....25..26.....

..|\\\./../.....\..\..\.....|\\\.../.../......

..|.\\X__/__...\..\..\....|.\\\/__/___....

..|..X\/__..\...\..\..\...|..\\/_/__..\...

..|./.\/..\..\...\..\..\..|...X___/_..\..\..

../.|..\..\..\..\...|..|..|.|../.../..\..\..\.

.27.28.29.30.31.32.33.34.35.36.37.38..39.40.41

(X means two crossing paths)

Conjecture: This features a certain structure (similar to the G-sequence A005206 or other sequences of this family: A163875 and A163873). If the (below) following two constructs (C and D) are added on top of their ends (either marked with C or D) one will (if starting with one instance of D) receive the above tree (x marks a node, o marks spaces for nodes that are not part of the construct but will be filled by the other construct):

Diagram of D:

......x...........

..../..\\\........

.../....\\.\......

..|......\\..\....

..|.......\.\..\..

..D..o.o...x.x..x.

...........|.|..|.

...........D.C..C.

(o will be filled by C)

Diagram of C:

\\...x.

\\\./..

.\\/...

../\\..

./.\\\.

C...\\\

(This means construct C crosses on its way from a(n) to n exactly three other paths, e.g. from 25 to 37)

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000

Programs

Mathematica

A163874[n_] := A163874[n] = If[n < 3, 0, n - A163874[A163874[n-3]]];
Array[A163874, 100, 0] (* Paolo Xausa, Jan 05 2025 *)