A192675 Floor-Sqrt transform of large Fine numbers (A000957).

0, 1, 0, 1, 1, 2, 4, 7, 13, 24, 46, 85, 160, 301, 570, 1083, 2064, 3943, 7553, 14501, 27901, 53784, 103859, 200867, 389044, 754502, 1465037, 2847895, 5541797, 10794360, 21044286, 41061688, 80182834, 156692019, 306417804, 599604941, 1174044166, 2300154199, 4508885393, 8843184248

Offset: 0

Author

Emanuele Munarini, Jul 07 2011

Keywords

• nonn

Crossrefs

Cf. A000957.

Programs

Maple

b:= proc(n) option remember; `if`(n<3, n*(2-n),
      ((7*n-12)*b(n-1)+(4*n-6)*b(n-2))/(2*n))
    end:
a:= n-> floor(sqrt(b(n))):
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 26 2023

Mathematica

FSFromSeries[f_,x_,n_] := Map[Floor[Sqrt[#]]&,CoefficientList[Series[f,{x,0,n}],x]]
FSFromSeries[(1+2x-Sqrt[1-4x])/(2x(2+x)),x,100]

Python

from math import isqrt
from itertools import count, islice
def A192675_gen(): # generator of terms
    yield from (0,1,0)
    a, c = 0, 1
    for n in count(1):
        yield isqrt(a:=(c:=c*((n<<2)+2)//(n+2))-a>>1)
A192675_list = list(islice(A192675_gen(),20)) # Chai Wah Wu, Apr 26 2023

Formula

a(n) = floor(sqrt(Fine(n))).

Extensions

a(0) inserted by Chai Wah Wu, Apr 26 2023