A176177 a(n) = 2*n*3^(n-1) - (3^n-1)/2.

0, 1, 8, 41, 176, 689, 2552, 9113, 31712, 108257, 364136, 1210505, 3985808, 13020305, 42249560, 136314617, 437641664, 1399018433, 4455335624, 14140847849, 44747066480, 141214768241, 444565011128, 1396457152601, 4377657815456, 13697832519329, 42788074776872, 133447955987273, 415595062931792, 1292538773705297, 4014877075845656

Offset: 0

Author

N. J. A. Sloane, Dec 18 2010

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• A. J. Guttmann, On the critical behavior of self-avoiding walks II, J. Phys. A 22 (1989), 2807-2813. See Table 1.
• J. Sondow and H. Yi, New Wallis- and Catalan-type infinite products for ..., arXiv:1005.2712 [math.NT], 2010.
• J. Sondow and H. Yi, New Wallis- and Catalan-type infinite products for ... , Amer. Math. Monthly, 117 (201), 912-917.
• Index entries for linear recurrences with constant coefficients, signature (7,-15,9).

Programs

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

Mathematica

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 *)
CoefficientList[Series[-x (1 + x)/((x - 1) (-1 + 3 x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 17 2014 *)

PARI

first(m)=vector(m,n,2*n*3^(n-1)-(3^n-1)/2); /* Anders Hellström, Aug 10 2015 */

Formula

G.f.: -x*(1+x) / ( (x-1)*(-1+3*x)^2 ). - R. J. Mathar, Sep 04 2013

a(n) = 7*a(n-1) - 15*a(n-2) + 9*a(n-3) for n>2. - Vincenzo Librandi, Jun 17 2014

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