A297890 -id:A297890 - cp's OEIS frontend

A291481 Number of terms of A293630 at stage n.

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

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)).

cp's OEIS Frontend

A291481 Number of terms of A293630 at stage n.

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

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