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

Author

Benoit Cloitre, Oct 15 2017

Keywords

• nonn

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

Examples

			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
   ...
		

Links

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

Crossrefs

Cf. A293630, A296564, A297890.

Programs

Mathematica

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

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

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)

Extensions

More terms from Andrey Zabolotskiy, Oct 15 2017