A322425 -id:A322425 - cp's OEIS frontend

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

Benoit Cloitre, Oct 14 2017

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.

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

"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.

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 *)

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];

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

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

A322423 Start with (1, 2, 3); when the sequence has n terms, extend it by appending a(n) copies of a(1..n-1). (Cf. "Look to the left" sequence A293630.)

Original entry on oeis.org

1, 2, 3, 1, 2, 1, 2, 1, 2, 1, 2, 3, 1, 2, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 1, 2, 1, 2, 3, 1, 2, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 1, 2, 3, 1, 2, 1, 2, 1, 2, 1, 2, 3, 1, 2, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 1, 2, 1, 2, 3, 1

Offset: 1

Iain Fox, Dec 07 2018

nonn

			Sequence starts with 1, 2, 3.  Then 3 copies of 1, 2 are appended, giving 1, 2, 3, 1, 2, 1, 2, 1, 2.  Then 2 copies of everything but the final entry 2 are appended, giving 9 + 8 + 8 + 8 = 33 terms.

Iain Fox, Table of n, a(n) for n = 1..10000

"Look to the left" sequences: A293630 (seed 1,2), A322424 (seed 1,2,3,4), A322425 (seed 1,2,3,4,5).

Cf. A322426.

Nest[Join[#, Flatten@ ConstantArray[Drop[#, -1], #[[-1]]] ] &, {1, 2, 3}, 4] (* Michael De Vlieger, Dec 08 2018 *)

gen(n, v=[1,2,3], w) = for(x=1, n, w=vector(#v-1, i, v[i]); for(y=1, v[#v], v=concat(v, w))); v

Lim_{n->infinity} (a(1) + a(2) + ... + a(n))/n = 1.64948851...

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

A322424 Start with (1, 2, 3, 4); when the sequence has n terms, extend it by appending a(n) copies of a(1..n-1). (Cf. "Look to the left" sequence A293630.)

Original entry on oeis.org

1, 2, 3, 4, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 4, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 4, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 1, 2, 3, 4, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 1, 2, 3

Offset: 1

Iain Fox, Dec 07 2018

nonn

			Sequence starts with 1, 2, 3, 4.  Then 4 copies of 1, 2, 3 are appended, giving 1, 2, 3, 4, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3.  Then 3 copies of everything but the final entry 3 are appended, giving 16 + 15 + 15 + 15 = 61 terms.

Iain Fox, Table of n, a(n) for n = 1..10000

"Look to the left" sequences: A293630 (seed 1,2), A322423 (seed 1,2,3), A322425 (seed 1,2,3,4,5).

Cf. A322426.

Nest[Join[#, Flatten@ ConstantArray[Drop[#, -1], #[[-1]]] ] &, Range@ 4, 3] (* Michael De Vlieger, Dec 08 2018 *)

gen(n, v=[1,2,3,4], w) = for(x=1, n, w=vector(#v-1, i, v[i]); for(y=1, v[#v], v=concat(v, w))); v

Lim_{n->infinity} (a(1) + a(2) + ... + a(n))/n = 2.08405776...

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

A322426 Array read by downwards antidiagonals: A(n, k) is the k-th term of a "Look to the left" sequence starting with 1,2,...,n (see A293630).

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 3, 2, 1, 2, 2, 4, 3, 2, 1, 1, 1, 1, 4, 3, 2, 1, 1, 2, 2, 5, 4, 3, 2, 1, 2, 1, 3, 1, 5, 4, 3, 2, 1, 1, 2, 1, 2, 6, 5, 4, 3, 2, 1, 1, 1, 2, 3, 1, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 2, 7, 6, 5, 4, 3, 2, 1, 2, 3, 1, 1, 3, 1, 7, 6, 5

Offset: 2

Iain Fox, Dec 07 2018

			| n\k |  1   2   3   4   5   6   7   8   9  10  11  12  13  ...
|-----|-----------------------------------------------------------
|  2  |  1,  2,  1,  1,  1,  2,  1,  1,  2,  1,  1,  1,  2, ...
|  3  |  1,  2,  3,  1,  2,  1,  2,  1,  2,  1,  2,  3,  1, ...
|  4  |  1,  2,  3,  4,  1,  2,  3,  1,  2,  3,  1,  2,  3, ...
|  5  |  1,  2,  3,  4,  5,  1,  2,  3,  4,  1,  2,  3,  4, ...
|  6  |  1,  2,  3,  4,  5,  6,  1,  2,  3,  4,  5,  1,  2, ...
|  7  |  1,  2,  3,  4,  5,  6,  7,  1,  2,  3,  4,  5,  6, ...
|  8  |  1,  2,  3,  4,  5,  6,  7,  8,  1,  2,  3,  4,  5, ...
| ... |

Iain Fox, Antidiagonals n = 1..141 of array, flattened

n-th row: A293630 (n=2), A322423 (n=3), A322424 (n=4), A322425 (n=5).

row(n, k) = my(v=vector(n, i, i), w); while(#v

If k <= n, A(n, k) = k.

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A322423 Start with (1, 2, 3); when the sequence has n terms, extend it by appending a(n) copies of a(1..n-1). (Cf. "Look to the left" sequence A293630.)

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A322424 Start with (1, 2, 3, 4); when the sequence has n terms, extend it by appending a(n) copies of a(1..n-1). (Cf. "Look to the left" sequence A293630.)

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A322426 Array read by downwards antidiagonals: A(n, k) is the k-th term of a "Look to the left" sequence starting with 1,2,...,n (see A293630).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula