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

0,3

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

Formula

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

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

Prog

(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); ); }

Crossrefs

Cf. A000045, A370730.

Sequence in context: A272591 A339694 A074722 * A331102 A080368 A057174

Adjacent sequences: A370741 A370742 A370743 * A370745 A370746 A370747

Keyword

nonn,base

Author

Rémy Sigrist, Feb 29 2024

Status

approved