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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 October 15 01:40 EDT 2019. Contains 328025 sequences. (Running on oeis4.)