A165310 a(0)=1, a(1)=3, a(n) = 7*a(n-1) - 9*a(n-2) for n > 1.

1, 3, 12, 57, 291, 1524, 8049, 42627, 225948, 1197993, 6352419, 33684996, 178623201, 947197443, 5022773292, 26634636057, 141237492771, 748950724884, 3971517639249, 21060066950787, 111676809902268, 592197066758793

Offset

0,2

Comments

a(n)/a(n-1) tends to (7+sqrt(13))/2 = 5.30277563... = 2+A098316.

For n >= 2, a(n) equals 3^n times the permanent of the (2n-2) X (2n-2) matrix with 1/sqrt(3)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Index entries for linear recurrences with constant coefficients, signature (7,-9).

Formula

G.f.: (1-4*x)/(1-7*x+9*x^2).

a(n) = Sum_{k=0..n} A165253(n,k)*3^(n-k).

a(n) = ((13-sqrt(13))*(7+sqrt(13))^n+(13+sqrt(13))*(7-sqrt(13))^n )/(26*2^n). - Klaus Brockhaus, Sep 26 2009

Mathematica

LinearRecurrence[{7, -9}, {1, 3}, 30] (* Harvey P. Dale, Sep 23 2011 *)
CoefficientList[Series[(1 - 4 x)/(1 - 7 x + 9 x^2), {x, 0, 30}], x] (* Wesley Ivan Hurt, Feb 12 2026 *)

Prog

(Magma) I:=[1, 3]; [n le 2 select I[n] else 7*Self(n-1)-9*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 24 2011
(PARI) a(n)=([0, 1; -9, 7]^n*[1; 3])[1, 1] \\ Charles R Greathouse IV, May 26 2026

Crossrefs

Cf. A098316, A165253.

Sequence in context: A101106 A389890 A389286 * A133158 A328295 A194089

Adjacent sequences: A165307 A165308 A165309 * A165311 A165312 A165313

Keyword

nonn,easy

Author

Philippe Deléham, Sep 14 2009

Status

approved