OFFSET
1,2
COMMENTS
If n is written in base-4 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m-2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).
FORMULA
Recurrence: a(n)=n*a(floor(n/4)); a(n*4^m)=n^m*4^(m(m+1)/2)*a(n).
a(k*4^m)=k^(m+1)*4^(m(m+1)/2), for 0<k<4.
Asymptotic behavior: a(n)=O(n^((1+log_4(n))/2)); this follows from the inequalities below.
a(n)<=b(n), where b(n)=n^(1+floor(log_4(n)))/4^((1+floor(log_4(n)))*floor(log_4(n))/2); equality holds for n=k*4^m, 0<k<4, m>=0. b(n) can also be written n^(1+floor(log_4(n)))/4^A000217(floor(log_4(n))).
Also: a(n)<=2^(1/4)*n^((1+log_4(n))/2)=1.189207...*4^A000217(log_4(n)), equality holds for n=2*4^m, m>=0.
a(n)>c*b(n), where c=0.4194224417951075977... (see constant A132020).
Also: a(n)>c*2^(1/4)*n^((1+log_4(n))/2)=0.498780...*4^A000217(log_4(n)).
lim inf a(n)/b(n)=0.4194224417951075977..., for n-->oo.
lim sup a(n)/b(n)=1, for n-->oo.
lim inf a(n)/n^((1+log_4(n))/2)=0.4194224417951075977...*2^(1/4), for n-->oo.
lim sup a(n)/n^((1+log_4(n))/2)=2^(1/4), for n-->oo.
lim inf a(n)/a(n+1)=0.4194224417951075977... for n-->oo (see constant A132020).
EXAMPLE
a(26)=floor(26/4^0)*floor(26/4^1)*floor(26/4^2)=26*6*1=156; a(34)=544 since 34=202(base-4) and so
a(34)=202*20*2(base-4)=34*8*2=544.
CROSSREFS
KEYWORD
nonn
AUTHOR
Hieronymus Fischer, Aug 20 2007
STATUS
approved