A296564 -id:A296564 - 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

A291481 Number of terms of A293630 at stage n.

Original entry on oeis.org

2, 4, 7, 13, 37, 73, 145, 289, 865, 1729, 3457, 10369, 20737, 41473, 82945, 248833, 497665, 995329, 1990657, 3981313, 11943937, 23887873, 47775745, 143327233, 286654465, 573308929, 1146617857, 3439853569, 6879707137, 20639121409, 41278242817, 82556485633

Offset: 0

Benoit Cloitre, Oct 15 2017

nonn

limsup a(n+1)/a(n) = 3, liminf a(n+1)/a(n) = 2 (n->oo). It seems that lim_{n->oo} a(n)^(1/n) = C with C > 2.
Limit_{n->oo} a(n)^(1/n) = 2.236151... (see A297890). - Jon E. Schoenfield, Dec 23 2017
The previous limit is also equal to 2^(2 - d) * 3^(d - 1), where d = 1.275261... (see A296564). - Iain Fox, Dec 24 2017

			A293630 at stage n:
  n = 0: [1, 2];                                      2 terms
  n = 1: [1, 2, 1, 1];                                4 terms
  n = 2: [1, 2, 1, 1, 1, 2, 1];                       7 terms
  n = 3: [1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2];    13 terms
  n = 4: [1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, ...];  37 terms
   ...

Iain Fox, Table of n, a(n) for n = 0..2860

Cf. A293630, A296564, A297890.

Length /@ NestList[Join[#, Join @@ ConstantArray[Most[#], Last[#]]] &, {1, 2}, 24] (* Michael De Vlieger, Jan 21 2018 *)

v=[1,2];for(n=1,17,l=length(v);w=vector(l-1,i,v[i]);v=concat(v,if(v[l]-1,concat(w,w),w));print1(length(v),","));

lista(nn) = {
my(S = [1, 2], n = 2, L, nPrev, E);
print1("2, ");
for(j = 1, nn, L = S[#S]; n = n*(1+L)-L; nPrev = #S; for(r = 1, L, for(i = 1, nPrev-1, S = concat(S, S[i]))); print1(n, ", "));
E = S;
for(j = nn + 1, nn + #E, L = E[#E+1-(j-nn)]; n = n*(1+L)-L; print1(n, ", "))
} \\ Iain Fox, Jan 21 2018

a, z = [1, 2], [2]
while z[-1]<1000:
    a += a[:-1]*a[-1]
    z.append(len(a))
for i in range(100):
    z.append((z[-1]-1)*(a.pop()+1)+1)
print(z)
# Andrey Zabolotskiy, Oct 15 2017

From Iain Fox, Jan 21 2018: (Start)
a(n) = (1 + A293630(a(n-1)))*a(n-1) - A293630(a(n-1)).
a(n) ~ c^n, where c = 2.236151... (see comments or A297890).
(End)

More terms from Andrey Zabolotskiy, Oct 15 2017

A297890 Decimal expansion of lim_{k->infinity} (A291481(k)^(1/k)).

Original entry on oeis.org

2, 2, 3, 6, 1, 5, 1, 3, 7, 7, 4, 6, 8, 7, 4, 9, 0, 3, 9, 6, 6, 2, 7, 7, 8, 4, 0, 6, 6, 1, 2, 0, 4, 4, 0, 7, 3, 9, 1, 1, 4, 6, 4, 9, 9, 2, 4, 0, 2, 3, 9, 5, 7, 5, 4, 6, 3, 2, 1, 2, 5, 5, 5, 2, 6, 6, 5, 7, 9, 7, 4, 3, 1, 9, 3, 6, 4, 0, 2, 8, 6, 2, 6, 9, 8, 1, 0

Offset: 1

Iain Fox, Jan 07 2018

Equals 2^(2 - d) * 3^(d - 1), where d = lim_{k->infinity} (1/k)*Sum_{i=1..k} A293630(i) = 1.275261... (see A296564).
See comments from Jon E. Schoenfield on A296564 for explanation of PARI program.
Is this number transcendental?

			Equals 2.2361513774687490396627784066120440739114649924...
Values evaluated with A291481(k):
  k = 1: 4^(1/1)                 = 4
  k = 2: 7^(1/2)                 = 2.645751311064590590...
  k = 3: 13^(1/3)                = 2.351334687720757489...
  k = 4: 37^(1/4)                = 2.466325714559660444...
  k = 5: 73^(1/5)                = 2.358655818240735626...
  k = 6: 145^(1/6)               = 2.292070651723655173...
  ...
  k = infinity: A291481(k)^(1/k) = 2.236151377468749039...

Iain Fox, Table of n, a(n) for n = 1..20000

Cf. A291481, A293630, A296564.

gen(build) = {
my(S = [1, 2], n = 2, t = 3, L, nPrev, E);
for(j = 1, build, L = S[#S]; n = n*(1+L)-L; t = t*(1+L)-L^2; nPrev = #S; for(r = 1, L, for(i = 1, nPrev-1, S = concat(S, S[i]))));
E = S;
for(j = build + 1, build + #E, L = E[#E+1-(j-build)]; n = n*(1+L)-L; t = t*(1+L)-L^2);
return(2^(2 - t/n)*3^(t/n - 1))
} \\ (gradually increase build to get more precise answers)

A297927 Decimal expansion of ratio of number of 1's to number of 2's in A293630.

Original entry on oeis.org

2, 6, 3, 2, 9, 0, 4, 5, 5, 5, 1, 7, 9, 0, 6, 5, 9, 4, 5, 7, 9, 8, 7, 2, 8, 5, 5, 6, 7, 5, 3, 5, 9, 7, 4, 5, 7, 1, 1, 5, 5, 7, 0, 6, 2, 9, 0, 9, 8, 6, 4, 2, 3, 8, 0, 2, 3, 2, 2, 2, 0, 3, 4, 7, 4, 9, 3, 2, 5, 9, 4, 7, 2, 2, 1, 3, 0, 6, 9, 1, 2, 1, 3, 5, 6, 1, 9

Offset: 1

Iain Fox, Jan 08 2018

Equals (2 - d)/(d - 1), where d = lim_{k->infinity} (1/k)*Sum_{i=1..k} A293630(i) = 1.275261... (see A296564).
See comments from Jon E. Schoenfield on A296564 for explanation of PARI program.
Is this number transcendental?

			Equals 2.6329045551790659457987285567535974571155706290...
After generating k steps of A293630:
  k = 0:        [1, 2];                  1
  k = 1:        [1, 2, 1, 1];            3
  k = 2:        [1, 2, 1, 1, 1, 2, 1];   2.5
  k = 3:        [1, 2, 1, 1, 1, 2, ...]; 2.25
  k = 4:        [1, 2, 1, 1, 1, 2, ...]; 2.7
  k = 5:        [1, 2, 1, 1, 1, 2, ...]; 2.65
  k = 6:        [1, 2, 1, 1, 1, 2, ...]; 2.625
  ...
  k = infinity: [1, 2, 1, 1, 1, 2, ...]; 2.632904555179...

Iain Fox, Table of n, a(n) for n = 1..20000

Cf. A293630, A296564.

gen(build) = {
my(S = [1, 2], n = 2, t = 3, L, nPrev, E);
for(j = 1, build, L = S[#S]; n = n*(1+L)-L; t = t*(1+L)-L^2; nPrev = #S; for(r = 1, L, for(i = 1, nPrev-1, S = concat(S, S[i]))));
E = S;
for(j = build + 1, build + #E, L = E[#E+1-(j-build)]; n = n*(1+L)-L; t = t*(1+L)-L^2);
return(1.0*(2 - t/n)/(t/n - 1))
} \\ (gradually increase build to get more precise answers)

Terms after a(3) corrected by Iain Fox, Jan 16 2018

A298590 Sum of terms of A293630 after generating the sequence for n steps (see comments).

Original entry on oeis.org

3, 5, 9, 17, 47, 93, 185, 369, 1103, 2205, 4409, 13223, 26445, 52889, 105777, 317327, 634653, 1269305, 2538609, 5077217, 15231647, 30463293, 60926585, 182779751, 365559501, 731119001, 1462238001, 4386713999, 8773427997, 26320283987, 52640567973, 105281135945

Offset: 0

Iain Fox, Jan 22 2018

nonn

A293630, without generating it, starts as 1, 2. After 1 step, the block to the left is repeated twice and results in 1, 2, 1, 1. Generating a second step gives 1, 2, 1, 1, 1, 2, 1. This continues and a(n) is the sum of the terms at the n-th step.
A291481(n) < a(n) < 2*A291481(n).
Lim_{k->infinity} a(k)/A291481(k) = 1.275261... (see A296564).
Lim_{k->infinity} a(k)^(1/k) = 2.236151... (see A297890).

			A293630 generated n times.
  n = 0: [1, 2];                   a(0) = 1 + 2 = 3.
  n = 1: [1, 2, 1, 1];             a(1) = 1 + 2 + 1 + 1 = 5.
  n = 2: [1, 2, 1, 1, 1, 2, 1];    a(2) = 1 + 2 + 1 + 1 + 1 + 2 + 1 = 9.
  n = 3: [1, 2, 1, 1, 1, 2, ...];  a(2) = 1 + 2 + 1 + 1 + 1 + 2 + ... = 17.
   ...

Iain Fox, Table of n, a(n) for n = 0..2860

Cf. A291481, A293630, A296564, A297890.

lista(nn) = {
my(S = [1, 2], t = 3, L, nPrev, E);
print1("3, ");
for(j = 1, nn, L = S[#S]; t = t*(1+L)-L^2; nPrev = #S; for(r = 1, L, for(i = 1, nPrev-1, S = concat(S, S[i]))); print1(t, ", "));
E = S;
for(j = nn + 1, nn + #E, L = E[#E+1-(j-nn)]; t = t*(1+L)-L^2; print1(t, ", "));
}

a(n) = Sum_{k=1..A291481(n)} A293630(k).
a(n) = (1 + A293630(A291481(n-1)))*a(n-1) - A293630(A291481(n-1))^2.
a(n) ~ d*A291481(n), where d = 1.275261... (see A296564).
a(n) = A298606(A291481(n)).

A298606 Partial sums of A293630.

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 17, 18, 20, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 35, 36, 37, 38, 40, 41, 42, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 56, 57, 59, 60, 61, 62, 64, 65, 67, 68, 69, 70, 72, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84, 85

Offset: 1

Iain Fox, Jan 22 2018

nonn

a(n) is n plus the number of 2's in A293630 on the interval 1..n.

			A293630 begins 1, 2, 1, 1, 1, 2, 1, 1, 2, ... so:
a(1) = 1.
a(2) = 1 + 2 = 3.
a(3) = 1 + 2 + 1 = 4.
a(4) = 1 + 2 + 1 + 1 = 5.
...

Iain Fox, Table of n, a(n) for n = 1..10369

Cf. A291481, A293630, A296564, A298590.

Accumulate@ Nest[Join @@ {#, Join @@ ConstantArray[Most@ #, Last@ #]} &, {1, 2}, 5] (* Michael De Vlieger, Jan 22 2018 *)

lista(nn) = my(v=[1, 2], l, w, s=0); for(n=1, nn, l=length(v); w=vector(l-1, i, v[i]); v=concat(v, if(v[l]-1, concat(w, w), w))); for(i=1, length(v), s += v[i]; print1(s, ", "))

a(A291481(n)) = A298590(n).
Lim_{k->infinity} a(k)/k = 1.275261... (see A296564).
a(n) ~ d*n, where d = 1.275261... (see A296564).

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).

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A291481 Number of terms of A293630 at stage n.

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A297890 Decimal expansion of lim_{k->infinity} (A291481(k)^(1/k)).

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A297927 Decimal expansion of ratio of number of 1's to number of 2's in A293630.

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A298590 Sum of terms of A293630 after generating the sequence for n steps (see comments).

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A298606 Partial sums of A293630.

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