A293630 "Look to the left" sequence starting with (1, 2): when the sequence has n terms, extend it by appending a(n) copies of a(1..n-1).

1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2

Offset: 1

Author

Benoit Cloitre, Oct 14 2017

Keywords

• nonn
• nice

Comments

Stage 1: last term of 1,2 is 2 hence we add 2 copies of the block to the left of the last term (here 1) giving 1,2,1,1.

Stage 2: last term of 1,2,1,1 is 1 hence we add one copy of the block to the left of the last term (here 1,2,1) giving 1,2,1,1,1,2,1.

Stage 3: last term of 1,2,1,1,1,2,1 is 1 hence we add one copy of the block to the left of the last term (here 1,2,1,1,1,2) giving 1,2,1,1,1,2,1,1,2,1,1,1,2.

Iterate the process.

Links

• Iain Fox, Table of n, a(n) for n = 1..10369
• Benoit Cloitre, Plot of Sum_{k=1..n} a(k)/n

Crossrefs

"Look to the left" sequences: A322423 (seed 1,2,3), A322424 (seed 1,2,3,4), A322425 (seed 1,2,3,4,5).

Cf. A291481, A296564, A297927, A322426.

Programs

Mathematica

f[s_List] := Block[{a = Flatten[s][[-1]], b = Most@ s}, s = Join[s, Flatten@ Table[b, {a}]]]; Nest[f, {1, 2}, 6] (* Robert G. Wilson v, Dec 23 2017 *)

PARI

v=[1,2];for(n=1,10,l=length(v);w=vector(l-1,i,v[i]);v=concat(v,if(v[l]-1,concat(w,w),w)));a(n)=v[n];

Formula

It seems that lim_{n->infinity} (a(1) + a(2) + ... + a(n))/n = 1.27526... (see link and A296564).

Because of the previous statement, it seems that the ratio of 2s to 1s in this sequence is 1:2.6329... (see A297927). - Iain Fox, Oct 15 2017

Extensions

Self-contained name from M. F. Hasler, Dec 10 2018