A112884 Number of bits required to represent binomial(2^n, 2^(n-1)).

2, 3, 7, 14, 30, 61, 125, 252, 508, 1019, 2043, 4090, 8186, 16377, 32761, 65528, 131064, 262135, 524279, 1048566, 2097142, 4194293, 8388597

Offset

1,1

Links

Table of n, a(n) for n=1..23.

Formula

Appears to be equal to 2^n - floor(n / 2).

G.f.: x*(-2*x^3 + 3*x - 2)/((x - 1)^2*(2*x^2 + x - 1)) [Conjectured by Harvey P. Dale, Apr 06 2011]

a(n) = A070939(A037293(n)). - Alois P. Heinz, Feb 17 2024

Example

a(2) = 3 because binomial(2^2, 2^1) in binary = 110.

Mathematica

Table[IntegerLength[Binomial[2^n, 2^(n-1)], 2], {n, 25}] (* or *)
CoefficientList[Series[(-2 x^3+3x-2)/((x-1)^2 (2x^2+x-1)), {x, 0, 25}], x] (* Harvey P. Dale, Apr 06 2011 *)

Prog

(PHP) $LastFact = gmp_init('1'); for ($i = 2; $i !== 65536; $i *= 2) { $Fact = gmp_fact($i); $Result = gmp_div_q($Fact, gmp_pow($OldFact, 2)); $LastFact = $Fact; echo gmp_strval($Result, 2).'<br>'; }

Crossrefs

a(n) represents the size of A037293 in binary - see also the central binomial coefficients: A001405.

Cf. A070939.

Sequence in context: A281716 A192570 A019595 * A103421 A205484 A151530

Adjacent sequences: A112881 A112882 A112883 * A112885 A112886 A112887

Keyword

easy,nonn

Author

Matt Erbst (matt(AT)erbst.org), Oct 04 2005

Status

approved