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A215928 a(n) = 2*a(n-1) + a(n-2) for n > 2, a(0) = a(1) = 1, a(2) = 2. 5

%I

%S 1,1,2,5,12,29,70,169,408,985,2378,5741,13860,33461,80782,195025,

%T 470832,1136689,2744210,6625109,15994428,38613965,93222358,225058681,

%U 543339720,1311738121,3166815962,7645370045,18457556052,44560482149,107578520350,259717522849

%N a(n) = 2*a(n-1) + a(n-2) for n > 2, a(0) = a(1) = 1, a(2) = 2.

%C Pell numbers with a(0) replaced by 1. - _R. J. Mathar_, Sep 08 2012

%C Number of 132-avoiding two-stack sortable permutations. See Theorem 2.2 of Egge and Mansour which gives a generating function equation P(x) = 1 + x + 2*x^2 + x*(P(x) - 1 - x) + x^2*(P(x) - 1) + x*(P(x) - 1 - x).

%C Row sums of triangle A155161. - _Philippe Deléham_, Aug 31 2012

%C a(n) is the top left entry of the n-th power of any of the 3 X 3 matrices [1, 1, 1; 1, 1, 1; 0, 1, 0] or [1, 1, 0; 1, 1, 1; 1, 1, 0] or [1, 1, 1; 0, 0, 1; 1, 1, 1] or [1, 0, 1; 1, 0, 1; 1, 1, 1]. - _R. J. Mathar_, Feb 03 2014

%C For n > 0, A001333(n)/a(n) = A001333(n)/A000129(n), which converges to sqrt(2). - _Karl V. Keller, Jr._, May 17 2015

%H Karl V. Keller, Jr., <a href="/A215928/b215928.txt">Table of n, a(n) for n = 0..500</a>

%H Phan Thuan Do, Thi Thu Huong Tran, Vincent Vajnovszki, <a href="https://arxiv.org/abs/1809.00742">Exhaustive generation for permutations avoiding a (colored) regular sets of patterns</a>, arXiv:1809.00742 [cs.DM], 2018.

%H E. S. Egge and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0205206">132-avoiding Two-stack Sortable Permutations, Fibonacci Numbers, and Pell Numbers</a>, arXiv:math/0205206 [math.CO], 2002.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2, 1).

%F a(n) = 2*a(n-1) + a(n-2) for n > 2, a(0) = a(1) = 1, a(2) = 2.

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

%F a(n) = A000129(n) unless n = 0.

%F a(n+1) - a(n) = A078057(n-1).

%F PSUM transform is A024537.

%F PSUMSIGN transform is A097075.

%F INVERT transform of A142474.

%F G.f.: 1/( 1 - (Sum_{k>=0} x*(x + x^2)^k) ) = 1/( 1 - (Sum_{k>=1} x/(1 - x^2))^k) ). - _Joerg Arndt_, Sep 30 2012

%F a(n) = 1/2*((1 + sqrt(2))^(n - 1) + (1 - sqrt(2))^(n - 1)) + 1/4*sqrt(2)*((1 + sqrt(2))^(n - 1) - (1 - sqrt(2))^(n - 1)), for n > 0. - _Paolo P. Lava_, Oct 26 2012

%F G.f.: 1 + Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k + 2 + x)/( x*(4*k + 4 + x) + 1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Sep 06 2013

%F a(n) = A069306(n-1) if n > 1. - _Michael Somos_, Oct 23 2018

%F E.g.f.: 1 + exp(x)*sinh(sqrt(2)*x)/sqrt(2). - _Franck Maminirina Ramaharo_, Nov 29 2018

%e G.f. = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 29*x^5 + 70*x^6 + 169*x^7 + 408*x^8 + 985*x^9 + ...

%p f:= gfun:-rectoproc({a(n)=2*a(n-1)+a(n-2), a(0)=1, a(1)=1, a(2)=2}, a(n), remember):

%p map(f, [$0..100]); # _Robert Israel_, May 29 2015

%t CoefficientList[Series[(1 - x - x^2)/(1 - 2 x - x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, May 14 2015 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( 1 / (1 - x / (1 - x / (1 - x / (1 + x)))) + x * O(x^n), n))};

%o (MAGMA) [1] cat [ n le 2 select (n) else 2*Self(n-1)+Self(n-2): n in [1..35] ]; // _Vincenzo Librandi_, May 14 2015

%Y Cf. A000129, A024537, A069306, A078057, A097075, A142474.

%K nonn,easy

%O 0,3

%A _Michael Somos_, Aug 27 2012

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Last modified April 10 05:04 EDT 2020. Contains 333392 sequences. (Running on oeis4.)