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 A099393 a(n) = 4^n + 2^n - 1. 15

%I

%S 1,5,19,71,271,1055,4159,16511,65791,262655,1049599,4196351,16781311,

%T 67117055,268451839,1073774591,4295032831,17180000255,68719738879,

%U 274878431231,1099512676351,4398048608255,17592190238719

%N a(n) = 4^n + 2^n - 1.

%C Number of occurrences of letter 2 in (n+1)-st Peano word.

%C 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_, Feb 07 2006

%C 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_, Aug 08 2007

%D A.M.Cohen,D.E. Taylor, American Math Monthly, volume 114,Number 7, Aug-Sept 2007, pages 633-638.

%H Vincenzo Librandi, <a href="/A099393/b099393.txt">Table of n, a(n) for n = 0..170</a>

%H S. Kitaev and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0210268">The Peano curve and counting occurrences of some pattern</a>

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,8).

%F a(n) = 2^(2*n-1) + 2*a(n-1) + 1. - _Roger L. Bagula_, Aug 08 2007

%F From _Mohammad K. Azarian_, Jan 15 2009: (Start)

%F G.f.: 1/(1-4*x) + 1/(1-2*x) - 1/(1-x).

%F E.g.f.: e^(4*x) + e^(2*x) - e^x. (End)

%F a(n) = A279396(n+4, 4). - _Wolfdieter Lang_, Jan 10 2017

%e n=5: a(5)=4^5+2^5-1=1024+32-1=1055 -> '10000011111'.

%t 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_, Aug 08 2007 *)

%t LinearRecurrence[{7,-14,8},{1,5,19},30] (* _Harvey P. Dale_, Sep 06 2015 *)

%o (MAGMA) [4^n + 2^n - 1: n in [0..60]]; // _Vincenzo Librandi_, Apr 26 2011

%o (PARI) a(n)=4^n+2^n-1 \\ _Charles R Greathouse IV_, Sep 24 2015

%Y Equals A063376(n) - 1.

%Y Cf. A279396.

%K nonn,easy

%O 0,2

%A _Ralf Stephan_, Oct 20 2004

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Last modified April 19 06:59 EDT 2019. Contains 322237 sequences. (Running on oeis4.)