A100665 a(n) = round(sqrt(Fibonacci(n))).

0, 1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 12, 15, 19, 25, 31, 40, 51, 65, 82, 105, 133, 169, 215, 274, 348, 443, 564, 717, 912, 1160, 1476, 1877, 2388, 3038, 3864, 4915, 6252, 7953, 10116, 12868, 16368, 20821, 26484, 33688, 42852, 54509, 69336, 88197, 112189, 142706

Offset

0,5

Links

Table of n, a(n) for n=0..51.

Formula

Also the rounded values of the sequence of real numbers g(n) defined by g(0) = 0, g(1) = 1, then g(n) = (g(n-1)^2 + g(n-2)^2)^(1/2). - Corrected by Thomas Ordowski, Jan 05 2013

This is a special case of the following conjectured relationship: Given a starting sequence of g(0)=0, g(1)=1, ..., g(m)=1 (i.e., 0 then m-1 1's) and then a recurrence relation g(n) = b_1*g(n-1)^c_1 + b_2*g(n-2)^c_2 + ... + b_m*g(n-m)^c_m where the b_i's and c_i's are real numbers > 0 then (b_1*(g(n-1)^c_1)^x + b_2*(g(n-2)^c_2)^x + ... + b_m*(g(n-m)^c_m)^x)^(1/x) = g(n)^(1/x) where x is a real number > 0. - Gerald McGarvey, Dec 12 2004

Mathematica

Table[ Round[ Sqrt[ Fibonacci[ n]]], {n, 0, 50}] (* Robert G. Wilson v Dec 10 2004 *)

Prog

(Python)
from gmpy2 import isqrt, fib
def A100665(n): return int((m:=isqrt(k:=fib(n)))+(k-m*(m+1)>=1)) # Chai Wah Wu, Jun 19 2024

Crossrefs

Cf. A000045.

Sequence in context: A199122 A086740 A120161 * A114095 A301513 A066639

Adjacent sequences: A100662 A100663 A100664 * A100666 A100667 A100668

Keyword

nonn

Author

Gerald McGarvey, Dec 04 2004

Extensions

More terms from Robert G. Wilson v, Dec 10 2004

Status

approved