%I A099393
%S A099393 1,5,19,71,271,1055,4159,16511,65791,262655,1049599,4196351,16781311,
%T A099393 67117055,268451839,1073774591,4295032831,17180000255,68719738879,
%U A099393 274878431231,1099512676351,4398048608255,17592190238719
%N A099393 4^n + 2^n - 1.
%C A099393 Number of occurrences of letter 2 in (n+1)-st Peano word.
%C A099393 a(n) = A020522(n)+A000225(n+1) = A083420(n)-A020522(n); in binary representation: 
               a leading one followed by n zeros then by n ones; A000120(a(n))=n+1; 
               A023416(a(n))=n; A070939(a(n))=2*n+1; 2*A020522(n)+1=A030101(a(n)). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 07 2006
%C A099393 The number of involutions in group G_n G_{n+1}=G_n(operation) D_8. For 
               example, Q_8->1 involution; D_8->5 involutions - Roger L. Bagula 
               (rlbagulatftn(AT)yahoo.com), Aug 08 2007
%D A099393 A.M.Cohen,D.E. Taylor, American Math Monthly, volume 114,Number 7, Aug-Sept 
               2007, pages 633-638
%H A099393 S. Kitaev and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0210268">
               The Peano curve and counting occurrences of some pattern</a>
%F A099393 a(n) = 2^(2*n-1)+2*a(n-1)+1 - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), 
               Aug 08 2007
%F A099393 G.f.: 1/(1-4*x)+1/(1-2*x)-1/(1-x). E.g.f.: e^(4*x)+e^(2*x)-e^x. [From 
               Mohammad K. Azarian (azarian(AT)evansville.edu), Jan 15 2009]
%e A099393 n=5: a(5)=4^5+2^5-1=1024+32-1=1055 -> '10000011111'.
%t A099393 f[n_Integer?Positive] := f[n] = 2^(2*(n - 1) + 1)+2*f[n - 1] + 1 f[0] 
               = 1; Table[f[n], {n, 0, 30}] - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), 
               Aug 08 2007
%Y A099393 Equals A063376(n) - 1.
%Y A099393 Sequence in context: A095073 A128349 A001834 this_sequence A083588 A149759 
               A149760
%Y A099393 Adjacent sequences: A099390 A099391 A099392 this_sequence A099394 A099395 
               A099396
%K A099393 nonn
%O A099393 0,2
%A A099393 Ralf Stephan, Oct 20 2004