A276788 First differences of A003144.

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

Offset: 1

Author

N. J. A. Sloane, Oct 14 2016

Keywords

• nonn
• easy

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

• Robert Israel, Table of n, a(n) for n = 1..10608
• 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.

Crossrefs

Cf. A003144, A003145, A003146, A080843, A276790, A276791, A275925, A305374, A140101.

See A278039 for partial sums.

Programs

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

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