

A005943


Factor complexity (number of subwords of length n) of the GolayRudinShapiro binary word A020987.
(Formerly M1116)


4



1, 2, 4, 8, 16, 24, 36, 46, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 408, 416, 424, 432, 440, 448, 456, 464, 472, 480, 488, 496, 504
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OFFSET

0,2


COMMENTS

Terms a(0)..a(13) were verified and terms a(14)..a(32) were computed using the first 2^32 terms of the GRS sequence. [Joerg Arndt, Jun 10 2012]
Terms a(0)..a(63) were computed using the first 2^36 terms of the GRS sequence, and are consistent with Arndt's conjectured g.f.  Sean A. Irvine, Oct 12 2016
Needs a bfile.  N. J. A. Sloane, Jun 04 2019


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Table of n, a(n) for n=0..64.
JeanPaul Allouche, The Number of Factors in a Paperfolding Sequence, Bulletin of the Australian Mathematical Society, volume 46, number 1, August 1992, pages 2332. Section 6 theorem 2, a(n) = P_{w_i}(n).
J.P. Allouche and J. Shallit, The ring of kregular sequences, Theoretical Computer Sci., 98 (1992), 163197.
Index entries for linear recurrences with constant coefficients, signature (2,1).


FORMULA

G.f.: (1+x^2+2*x^3+4*x^4+4*x^62*x^72*x^9)/(1x)^2.  Joerg Arndt, Jun 10 2012
From Kevin Ryde, Aug 18 2020: (Start)
a(1..7) = 2,4,8,16,24,36,46, then a(n) = 8*n  8 for n>=8. [Allouche]
a(n) = 2*A337120(n1) for n>=1. [Allouche, end of proof of theorem 2]
(End)


EXAMPLE

All 8 subwords of length three (000, 001, ..., 111) occur in A020987, so a(3) = 8.


MAPLE

# Naive Maple program, useful for getting initial terms of factor complexity FC of a sequence b1[]. N. J. A. Sloane, Jun 04 2019
FC:=[0]; # a(0)=0 from the empty subword
for L from 1 to 12 do
lis := {};
for n from 1 to nops(b1)L do
s:=[seq(b1[i], i=n..n+L1)];
lis:={op(lis), s}; od:
FC:=[op(FC), nops(lis)];
od:
FC;


CROSSREFS

Cf. A006697, A005942, A337120 (paperfolding).
Sequence in context: A318654 A333994 A305656 * A330131 A008233 A224815
Adjacent sequences: A005940 A005941 A005942 * A005944 A005945 A005946


KEYWORD

nonn,nice


AUTHOR

N. J. A. Sloane, Jeffrey Shallit.


EXTENSIONS

Minor edits by N. J. A. Sloane, Jun 06 2012
a(14)a(32) added by Joerg Arndt, Jun 10 2012
a(33)a(36) added by Joerg Arndt, Oct 28 2012


STATUS

approved



