|
%I #40 Sep 08 2022 08:45:52
%S 0,1,8,41,176,689,2552,9113,31712,108257,364136,1210505,3985808,
%T 13020305,42249560,136314617,437641664,1399018433,4455335624,
%U 14140847849,44747066480,141214768241,444565011128,1396457152601,4377657815456,13697832519329,42788074776872,133447955987273,415595062931792,1292538773705297,4014877075845656
%N a(n) = 2*n*3^(n-1) - (3^n-1)/2.
%H Vincenzo Librandi, <a href="/A176177/b176177.txt">Table of n, a(n) for n = 0..1000</a>
%H A. J. Guttmann, <a href="http://dx.doi.org/10.1088/0305-4470/22/14/027">On the critical behavior of self-avoiding walks II</a>, J. Phys. A 22 (1989), 2807-2813. See Table 1.
%H J. Sondow and H. Yi, <a href="http://arxiv.org/abs/1005.2712">New Wallis- and Catalan-type infinite products for ...</a>, arXiv:1005.2712 [math.NT], 2010.
%H J. Sondow and H. Yi, <a href="http://www.jstor.org/stable/10.4169/000298910X523399">New Wallis- and Catalan-type infinite products for ... </a>, Amer. Math. Monthly, 117 (201), 912-917.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-15,9).
%F G.f.: -x*(1+x) / ( (x-1)*(-1+3*x)^2 ). - _R. J. Mathar_, Sep 04 2013
%F a(n) = 7*a(n-1) - 15*a(n-2) + 9*a(n-3) for n>2. - _Vincenzo Librandi_, Jun 17 2014
%F The companion matrix of the polynomial x^3 - 7*x^2 + 15*x - 9 is [(1,0,0); (1,3,0); (1,4,3)] = M, then M^n * [1,1,1] generates the sequence, extracting the lower term. - _Gary W. Adamson_, Aug 10 2015
%t Table[2n 3^(n-1)-(3^n-1)/2,{n,0,40}] (* or *) LinearRecurrence[{7,-15,9},{0,1,8},40] (* _Harvey P. Dale_, Jun 16 2014 *)
%t CoefficientList[Series[-x (1 + x)/((x - 1) (-1 + 3 x)^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Jun 17 2014 *)
%o (Magma) I:=[0,1,8]; [n le 3 select I[n] else 7*Self(n-1)-15*Self(n-2)+9*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Jun 17 2014
%o (PARI) first(m)=vector(m,n,2*n*3^(n-1)-(3^n-1)/2); /* _Anders Hellström_, Aug 10 2015 */
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Dec 18 2010
|
