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A291481
Number of terms of A293630 at stage n.
6
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
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
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Oct 15 2017
EXTENSIONS
More terms from Andrey Zabolotskiy, Oct 15 2017
STATUS
approved