A184327 a(1)=1, a(2)=17; thereafter a(n) = 6*a(n-1)-a(n-2)+c, where c=-4 if n is odd, c=12 if n is even.

1, 17, 97, 577, 3361, 19601, 114241, 665857, 3880897, 22619537, 131836321, 768398401, 4478554081, 26102926097, 152139002497, 886731088897, 5168247530881, 30122754096401, 175568277047521, 1023286908188737, 5964153172084897, 34761632124320657

Offset: 1

Author

N. J. A. Sloane, Dec 23 2012

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• J. V. Leyendekkers and A. G. Shannon, Pellian sequence relationships among pi, e, sqrt(2), Notes on Number Theory and Discrete Mathematics, Vol. 18, 2012, No. 2, 58-62. See Table 3, {y_n}.
• Index entries for linear recurrences with constant coefficients, signature (6,0,-6,1).

Programs

Magma

/* By definition: */ a:=[1,17]; c:=func; [n le 2 select a[n] else 6*Self(n-1)-Self(n-2)+c(n): n in [1..22]]; // Bruno Berselli, Dec 26 2012

Mathematica

CoefficientList[Series[(1 + 11 x - 5 x^2 + x^3)/((1 - x) (1 + x) (1 - 6 x + x^2)), {x, 0, 24}], x] (* Bruno Berselli, Dec 26 2012 *)

Formula

From Bruno Berselli, Dec 26 2012: (Start)

G.f.: x*(1+11*x-5*x^2+x^3)/((1-x)*(1+x)*(1-6*x+x^2)).

a(n) = a(-n) = 6*a(n-1)-6*a(n-3)+a(n-4).

a(n) = ((1+sqrt(2))^(2n)+(1-sqrt(2))^(2n))/2+(-1)^n-1.

a(n) = 2*A090390(n)-1. (End)

Extensions

Edited from Bruno Berselli, Dec 26 2012