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
KEYWORD
nonn,nice,easy
AUTHOR
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