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A011373
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Number of 1's in binary expansion of Fibonacci(n).
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10
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0, 1, 1, 1, 2, 2, 1, 3, 3, 2, 5, 4, 2, 5, 6, 4, 8, 7, 4, 5, 8, 6, 8, 11, 6, 6, 9, 11, 11, 12, 8, 11, 9, 13, 12, 11, 12, 14, 10, 12, 16, 17, 14, 16, 18, 15, 21, 13, 12, 18, 18, 17, 17, 17, 16, 22, 21, 16, 24, 20, 16, 19, 26, 23, 20, 25, 19, 26, 15, 23, 23, 22, 25, 27, 24, 23, 23, 22
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listen;
history;
text;
internal format)
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OFFSET
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0,5
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LINKS
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FORMULA
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a(n) = [x^Fibonacci(n)] (1/(1 - x))*Sum_{k>=0} x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Mar 27 2018
Conjecture: Limit_{n->oo} a(n)/n = log_2(phi)/2 = A242208/2 = 0.3471209568... . - Amiram Eldar, May 13 2022
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EXAMPLE
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a(8) = 3 because Fibonacci(8) = 21, which in binary is 11001 and that has 3 on bits.
a(9) = 2 because Fibonacci(9) = 34, which in binary is 100010 and that only has 2 on bits.
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MAPLE
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A000120 := proc(n) add(d, d=convert(n, base, 2)) ; end proc:
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MATHEMATICA
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DigitCount[#, 2, 1]&/@Fibonacci[Range[0, 79]] (* Harvey P. Dale, Mar 14 2011 *)
Table[Plus@@IntegerDigits[Fibonacci[n], 2], {n, 0, 79}]
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PROG
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(Scala) def fibonacci(n: BigInt): BigInt = {
val zero = BigInt(0)
def fibTail(n: BigInt, a: BigInt, b: BigInt): BigInt = n match {
case `zero` => a
case _ => fibTail(n - 1, b, a + b)
}
fibTail(n, 0, 1)
} // Based on tail recursion by Dario Carrasquel
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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