A005943 Factor complexity (number of subwords of length n) of the Golay-Rudin-Shapiro binary word A020987. (Formerly M1116)

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

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

References

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

Links

David A. Corneth, Table of n, a(n) for n = 0..9999

Jean-Paul Allouche, The Number of Factors in a Paperfolding Sequence, Bulletin of the Australian Mathematical Society, volume 46, number 1, August 1992, pages 23-32. Section 6 theorem 2, a(n) = P_{w_i}(n).

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.

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^6-2*x^7-2*x^9)/(1-x)^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(n-1) 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+L-1)];
    lis:={op(lis), s}; od:
FC:=[op(FC), nops(lis)];
od:
FC;

Mathematica

CoefficientList[Series[(1 + x^2 + 2 x^3 + 4 x^4 + 4 x^6 - 2 x^7 - 2 x^9)/(1 - x)^2, {x, 0, 64}], x] (* Michael De Vlieger, Oct 14 2021 *)

Prog

(PARI) first(n) = n = max(n, 10); concat([1, 2, 4, 8, 16, 24, 36, 46], vector(n-8, i, 8*i+48)) \\ David A. Corneth, Apr 28 2021

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,easy

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