cp's OEIS Frontend

A126269 Numbers n such that hcl(n,n) < hcl(n,n-1) where hcl(n,i) is the Huffman code length; see comments.

%I A126269 #10 Aug 06 2019 21:42:26
%S A126269 3,4,9,10,21,22,45,46,93,94,189,190,381,382,765,766,1533,1534,3069,
%T A126269 3070,6141,6142,12285,12286,24573,24574,49149,49150
%N A126269 Numbers n such that hcl(n,n) < hcl(n,n-1) where hcl(n,i) is the Huffman code length; see comments.
%C A126269 Consider a string which consists of n distinct symbols such that symbol(i) has frequency i (i=1,2,...,n). Then hcl(n,i) is the Huffman code length of symbol(i).
%H A126269 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a126/A126269.java">Java program</a> (github)
%F A126269 Conjecture: a(2k) = A033484(k-1) and a(2k-1) = A068156(k-1), k >= 2.
%F A126269 Conjectures from _Colin Barker_, Aug 06 2019: (Start)
%F A126269 G.f.: x^3*(3 + 4*x - 2*x^3) / ((1 - x)*(1 + x)*(1 - 2*x^2)).
%F A126269 a(n) = 3*a(n-2) - 2*a(n-4) for n>6.
%F A126269 a(n) = -5/2 + (-1)^n/2 + 3*2^((1/2)*(n-5))*(2-2*(-1)^n + sqrt(2) + (-1)^n*sqrt(2)) for n>2.
%F A126269 (End)
%e A126269 Possible Huffman codes for n = 3,4,5 are:
%e A126269 1 : 00
%e A126269 2 : 01
%e A126269 3 : 1
%e A126269 --------
%e A126269 1 : 100
%e A126269 2 : 101
%e A126269 3 : 11
%e A126269 4 : 0
%e A126269 --------
%e A126269 1 : 000
%e A126269 2 : 001
%e A126269 3 : 01
%e A126269 4 : 10
%e A126269 5 : 11
%e A126269 hcl(3,3)=1 < 2=hcl(3,2) and hcl(4,4)=1 < 2=hcl(4,3); so 3,4 are in the sequence.
%e A126269 hcl(5,5)=2=hcl(5,4) so 5 is not in the sequence.
%Y A126269 Cf. A126268 A126014 A068156 A033484.
%K A126269 easy,nonn
%O A126269 3,1
%A A126269 Serhat Sevki Dincer (mesti_mudam(AT)yahoo.com), Dec 22 2006
%E A126269 More terms from _Sean A. Irvine_, Aug 05 2019