A175395 a(n) = 2*Fibonacci(n)^2.

0, 2, 2, 8, 18, 50, 128, 338, 882, 2312, 6050, 15842, 41472, 108578, 284258, 744200, 1948338, 5100818, 13354112, 34961522, 91530450, 239629832, 627359042, 1642447298, 4299982848, 11257501250, 29472520898, 77160061448, 202007663442, 528862928882, 1384581123200, 3624880440722, 9490060198962, 24845300156168, 65045840269538, 170292220652450, 445830821687808, 1167200244410978, 3055769911545122, 8000109490224392, 20944558559128050

Offset: 0

Author

N. J. A. Sloane, Dec 03 2010

Keywords

• nonn
• easy

Comments

a(n) (n=1..) is half the number of nX2 binary arrays with no element equal to a strict majority of its diagonal and antidiagonal neighbors. - R. H. Hardin, Dec 02 2010

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Crossrefs

Cf. A000045, A007598, A236428.

Programs

Magma

[2*Fibonacci(n)^2: n in [0..50]]; // Vincenzo Librandi, Apr 24 2011

Mathematica

Table[2 Fibonacci[n]^2, {n, 0, 40}] (* Bruno Berselli, Nov 03 2015 *)
LinearRecurrence[{2,2,-1},{0,2,2},50] (* Harvey P. Dale, May 24 2023 *)

PARI

a(n) = round(2*(-2*(-1)^n+(1/2*(3-sqrt(5)))^n+(1/2*(3+sqrt(5)))^n)/5) \\ Colin Barker, Sep 28 2016

PARI

Vec(2*x*(1-x)/(1+x)/(1-3*x+x^2) + O(x^30)) \\ Colin Barker, Sep 28 2016

Formula

a(n) = 2*A007598(n).

G.f.: 2*x*(1-x)/(1+x)/(1-3*x+x^2). - Colin Barker, Feb 23 2012

a(n) = F(n-1)*F(n+1) + F(n-2)*F(n+2), where F = A000045, -F(-2) = F(-1) = 1. - Bruno Berselli, Nov 03 2015

a(n) = 2*(-2*(-1)^n+(1/2*(3-sqrt(5)))^n+(1/2*(3+sqrt(5)))^n)/5. - Colin Barker, Sep 28 2016

For n>1 a(n) is the denominator of the continued fraction [1, 1, ... 1, 2, 1, 1, ... 1, 2] with n-2 1's before each 2. See A236428 for the numerator. - Greg Dresden and Kevin Zhanming Zheng, Aug 16 2020