A370744 a(n) is the greatest Fibonacci number f such that f AND n = f (where AND denotes the bitwise AND operator).

0, 1, 2, 3, 0, 5, 2, 5, 8, 8, 8, 8, 8, 13, 8, 13, 0, 1, 2, 3, 0, 21, 2, 21, 8, 8, 8, 8, 8, 21, 8, 21, 0, 1, 34, 34, 0, 5, 34, 34, 8, 8, 34, 34, 8, 13, 34, 34, 0, 1, 34, 34, 0, 21, 34, 55, 8, 8, 34, 34, 8, 21, 34, 55, 0, 1, 2, 3, 0, 5, 2, 5, 8, 8, 8, 8, 8, 13

Offset: 0

Author

Rémy Sigrist, Feb 29 2024

Keywords

• nonn
• base

Comments

From Robert Israel, Mar 01 2024: (Start)

a(n) is the greatest Fibonacci number f <= n such that there are no carries in the base-2 addition of f and n-f.

a(n) is the greatest Fibonacci number f such that binomial(n, f) is odd. (End)

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000045, A370730.

Programs

Maple

Fib:= [seq(combinat:-fibonacci(n),n=0..100)]:
f:= proc(n) local m,k;
  m:= ListTools:-BinaryPlace(Fib,n+1);
  for k from m by -1 do
    if MmaTranslator:-Mma:-BitAnd(Fib[k],n) = Fib[k] then return Fib[k] fi
  od
end proc:
map(f, [$0..100]); # Robert Israel, Mar 01 2024

PARI

a(n) = { my (v = 0, f); for (k = 2, oo, f = fibonacci(k); if (f > n, return (v), bitand(f, n)==f, v = f);); }

Formula

a(n) <= n with equality iff n is a Fibonacci number.