cp's OEIS Frontend

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

%I A112884 #37 Oct 11 2024 13:08:08
%S A112884 1,2,3,7,14,30,61,125,252,508,1019,2043,4090,8186,16377,32761,65528,
%T A112884 131064,262135,524279,1048566,2097142,4194293,8388597,16777204,
%U A112884 33554420,67108851,134217715,268435442,536870898,1073741809,2147483633,4294967280,8589934576,17179869167
%N A112884 Number of bits required to represent binomial(2^n, 2^(n-1)).
%H A112884 Alois P. Heinz, <a href="/A112884/b112884.txt">Table of n, a(n) for n = 0..3321</a>
%H A112884 Sela Fried, <a href="/A112884/a112884.pdf">The number of bits required to represent binomial(2^n, 2^(n-1))</a>, 2024.
%H A112884 Sela Fried, <a href="https://arxiv.org/abs/2410.07237">Proofs of some Conjectures from the OEIS</a>, arXiv:2410.07237 [math.NT], 6 October 2024.
%H A112884 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,2).
%F A112884 Appears to be equal to 2^n - floor(n/2) = A000079(n) - A004526(n).
%F A112884 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
%F A112884 a(n) = A070939(A037293(n)). - _Alois P. Heinz_, Feb 17 2024
%F A112884 The conjectured formula 2^n - floor(n/2) and consequent g.f. are true (see links). - _Sela Fried_, Oct 03 2024
%e A112884 a(2) = 3 because binomial(2^2, 2^1) in binary = 110.
%t A112884 Table[IntegerLength[Binomial[2^n,2^(n-1)],2],{n,25}] (* or *)
%t A112884 CoefficientList[Series[(-2 x^3+3x-2)/((x-1)^2 (2x^2+x-1)), {x,0,25}], x] (* _Harvey P. Dale_, Apr 06 2011 *)
%o A112884 (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>'; }
%Y A112884 a(n) represents the size of A037293 in binary - see also the central binomial coefficients: A001405.
%Y A112884 Cf. A000079, A004526, A070939.
%K A112884 easy,nonn
%O A112884 0,2
%A A112884 Matt Erbst (matt(AT)erbst.org), Oct 04 2005
%E A112884 a(0)=1 prepended and g.f. adapted by _Alois P. Heinz_, Oct 11 2024