OFFSET
0,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..3321
Sela Fried, The number of bits required to represent binomial(2^n, 2^(n-1)), 2024.
Sela Fried, Proofs of some Conjectures from the OEIS, arXiv:2410.07237 [math.NT], 6 October 2024.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).
FORMULA
G.f.: (-3*x^3 + 2*x^2 + x - 1)/((x - 1)^2*(2*x^2 + x - 1)). - Conjectured by Harvey P. Dale, Apr 06 2011
The conjectured formula 2^n - floor(n/2) and consequent g.f. are true (see links). - Sela Fried, Oct 03 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
KEYWORD
easy,nonn
AUTHOR
Matt Erbst (matt(AT)erbst.org), Oct 04 2005
EXTENSIONS
a(0)=1 prepended and g.f. adapted by Alois P. Heinz, Oct 11 2024
STATUS
approved