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A276788 First differences of A003144. 8
2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
In A276790, leave 2's unchanged, but replace 1's by 2's and 0's by 1's, and then omit the initial 1.
If we prefixed A003144 with an initial 0, then its first differences would be a' := 1 followed by a, that is, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, ... If we now add 1 to every term of a' we get A305374 = first differences of A140101. - N. J. A. Sloane, Jul 17 2018
This relation between A003144 and A140101 is a conjecture - Michel Dekking, Mar 18 2019 [It has been a theorem since Mar 22 2019. - N. J. A. Sloane, Jun 25 2019. (See the Dekking et al. paper)]
(a(n)) is a morphic sequence: in the tribonacci word A092782 = 1,2,1,3,1,2,1,1,... map 1 -> 2, 2 -> 2, 3 -> 1. - Michel Dekking, Mar 21 2019
LINKS
Elena Barcucci, Luc Belanger and Srecko Brlek, On tribonacci sequences, Fib. Q., 42 (2004), 314-320. See page 316.
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
FORMULA
a(n) = A003144(n+1) - A003144(n), n >= 1.
a(n+1) = 2 - t(n)*(t(n) - 1)/2 = 2 - A276791(n+1), for n >= 0, where t(n) = A080843(n). See the W. Lang link in A080843, eq. (38). - Wolfdieter Lang, Dec 06 2018
MAPLE
M:= 10: # to use M generations of strings
S[1]:="a": S[2]:="ab": S[3]:="abac":
for n from 4 to M do S[n]:=cat(S[n-1], S[n-2], S[n-3]); od:
P:=select(t -> S[M][t]="a", [$1..length(S[M])]):
P[2..-1]-P[1..-2]; # Robert Israel, Nov 01 2016
CROSSREFS
See A278039 for partial sums.
Sequence in context: A076881 A355813 A136754 * A282625 A026498 A140685
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Oct 14 2016
STATUS
approved

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Last modified March 28 14:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)