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

0,1

Comments

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

Links

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

Formula

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)

Example

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
   ...

Mathematica

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

Prog

(PARI) 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), ", "));
(PARI) 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
(Python)
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

Crossrefs

Cf. A293630, A296564, A297890.

Sequence in context: A103104 A103480 A024826 * A102114 A102115 A102116

Adjacent sequences: A291478 A291479 A291480 * A291482 A291483 A291484

Keyword

nonn

Author

Benoit Cloitre, Oct 15 2017

Extensions

More terms from Andrey Zabolotskiy, Oct 15 2017

Status

approved