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A099393 a(n) = 4^n + 2^n - 1. 15
1, 5, 19, 71, 271, 1055, 4159, 16511, 65791, 262655, 1049599, 4196351, 16781311, 67117055, 268451839, 1073774591, 4295032831, 17180000255, 68719738879, 274878431231, 1099512676351, 4398048608255, 17592190238719 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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

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

REFERENCES

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..170

S. Kitaev and T. Mansour, The Peano curve and counting occurrences of some pattern

Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

FORMULA

a(n) = 2^(2*n-1) + 2*a(n-1) + 1. - Roger L. Bagula, Aug 08 2007

From Mohammad K. Azarian, Jan 15 2009: (Start)

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

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

a(n) = A279396(n+4, 4). - Wolfdieter Lang, Jan 10 2017

EXAMPLE

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

MATHEMATICA

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 *)

LinearRecurrence[{7, -14, 8}, {1, 5, 19}, 30] (* Harvey P. Dale, Sep 06 2015 *)

PROG

(MAGMA) [4^n + 2^n - 1: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

(PARI) a(n)=4^n+2^n-1 \\ Charles R Greathouse IV, Sep 24 2015

CROSSREFS

Equals A063376(n) - 1.

Cf. A279396.

Sequence in context: A255449 A296630 A001834 * A083588 A149759 A149760

Adjacent sequences:  A099390 A099391 A099392 * A099394 A099395 A099396

KEYWORD

nonn,easy

AUTHOR

Ralf Stephan, Oct 20 2004

STATUS

approved

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Last modified March 19 18:49 EDT 2019. Contains 321330 sequences. (Running on oeis4.)