A163875 a(n)=n-a(a(n-4)) with a(0)=a(1)=a(2)=a(3)=0.

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

Offset: 0

Author

Daniel Platt (d.platt(AT)web.de), Aug 08 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=4) - for general information about that family see A163873) Every a(n) occurs either exactly one or exactly five times (except from the initial values). A block of five occurrences of the same number n is after the first one interrupted by the following three elements: n+1,n+2 and n+3 (e.g. see from a(16) to a(23): 12, 13, 14, 15, 12, 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 55 elements the following (quintary) tree:

..............................4...................

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

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

......................8.......9.......10..11......

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

..................12.........13.........14...15...

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

................/...\_\\\_/........../.../......

.............../.../..\_\\\_______/.../.......

............./..../../._\_\\\________/........

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

.........16.....17.18.19..20.21.22.23.............

......../\\\\/__//__...\..\..\..\.............

......./..\\\/_/__/_..\...\..\..\..\............

....../....\\//__/_.\..\...\...\..\..\..........

...../......X__//_.\.\..\...\...\..\..\.........

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

...24....25.26.27..28.29.30.31.32....33.34.35.....

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

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

|....\\//__/.\..\..\.\.|./..|.\X_/__/.\.\...

|.....X__//_.\.\..\..\.\\/...|./\|_|.\.\.\..

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

36.27.38.39..40.41.42.43..44..48.49.50.51.52.\54.\

...........................45................53.55

...........................46.....................

...........................47.....................

(X means two crossing paths)

Conjecture: This features a certain structure (similar to the G-sequence A005206 or other sequences of this family: A163874 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.o..x.x.x.x.

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

............D.C.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 four other paths, e.g. from 18 to 26)

Links

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

Programs

Mathematica

A163875[n_] := A163875[n] = If[n < 4, 0, n - A163875[A163875[n-4]]];
Array[A163875, 100, 0] (* Paolo Xausa, Jan 05 2025 *)

Extensions

Terrible typos here and in A163874 and A163873! Corrected the sequence definition. Two further changes will be requested soon. A thousand apologies for the inconvenience Daniel Platt (d.platt(AT)web.de), Sep 14 2009