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A165565 a(n) = n-a(a(a(n-3))) with a(0)=a(1)=a(2)=0. 0
0, 0, 0, 3, 4, 5, 3, 3, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 12, 12, 12, 15, 16, 17, 15, 15, 15, 18, 19, 20, 21, 22, 23, 21, 21, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 30, 30, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41, 39, 39, 39, 42, 43, 44, 42, 42, 42, 45, 46, 47, 48, 49, 50, 51, 52 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A generalization of the Hofstadter G-sequence A005206 since it is part of the following family of sequences (which would give for k=1, p=2 the original G-sequence):

a(n)=n-(a^p)(n-k) where (a^p) denotes p recurrences of a on the given argument (e.g. this sequence would be denoted as n-(a^3)(n-3)) with a(0)=a(1)=...=a(k-1)=0 with k=1,2,3..., p=1,2,3... (here k=p=3)

Shares nearly all properties with the a(n)=n-(a(a(n-k)) family (quote from the page of the k=2-Sequence of this family, A163873, which applies to this family as well):

"Some things can be said about this family of sequences: Every a(n) occurs either exactly one or exactly k+1 times (except from the initial values which occur k times). A block of k+1 occurences of the same number n is after the first one interrupted by the following k-1 elements: n+1, n+2, ..., n+k-1 (e.g. see from [for this sequence: a(15) to a(20): 12,13,14,12,12,12]).

Since every natural number occurs in each sequence of the family at least once and 0<=a(n)<=n for all n [to be precisely: From the (2*k)-th element on] 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(n)

..|..

..a.."

This will give a (k+1)-ary tree which (Conjecture:) features a certain structure (similar to the G-sequence A005206 or other sequences of the above mentioned family:

A163873, A163875 and A163874). 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, constructs apply only to this sequence, comments for the whole family!):

Diagram of D:

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

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

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

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

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

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

...........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 k other paths, e.g. from 14 to 17)

The first node of D has always k+1 children nodes where the first one consists of a new copy of D, the second one consists of (p-1) other nodes and then D. The remaining children nodes consist of (p-1) other nodes and then C. Between the first and the second leaf is always space for k-1 nodes of construct C. Construct C crosses on its way from a(n) to n always exactly k paths (the right ones from construct D).

LINKS

Table of n, a(n) for n=0..73.

CROSSREFS

Mix from Hofstader H-Sequence A005374 and the Meta-Hofstadter G(-3)-Sequence A163874.

Sequence in context: A016553 A228947 A163874 * A033706 A121890 A178231

Adjacent sequences:  A165562 A165563 A165564 * A165566 A165567 A165568

KEYWORD

easy,nonn

AUTHOR

Daniel Platt (d.platt(AT)web.de), Sep 22 2009

STATUS

approved

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Last modified October 18 23:39 EDT 2019. Contains 328211 sequences. (Running on oeis4.)