It's still 2014!

Let $\alpha = c , \beta = d$

$c^{2} - 6c - 2 =0$

$c^{10} - 6c^{9} - 2c^{8} = 0$

similarly,

$d^{10} - 6d^{9} - 2d^{8} = 0$

Subtracting,

$c^{10} - d^{10} - 6( c^{9} - b^{9}) - 2(c^{8} - d^{8}) = 0$

$a_{10} - 2a_{8} = 6a_{9}$

$\frac{a_{10} - 2a_{8}}{2a_{9}} = 3$

$3^{942} = 2014k + 729$

hence remainder is 729

Since $\alpha$ and $\beta$ are roots of the equation: $\alpha \beta =-2\\ \alpha +\beta =6$

Now, multiplying and dividing by 3 and substituting the above values in the given expression, we get: $k=\frac { ({ (\alpha }^{ 10 }-{ \beta }^{ 10 })-2({ \alpha }^{ 8 }{ -\beta }^{ 8 })) }{ 2({ \alpha }^{ 9 }{ -\beta }^{ 9 }) } \times \frac { 3 }{ 3 } =\frac { 3({ \alpha }^{ 10 }-{ \beta }^{ 10 }+\alpha \beta ({ \alpha }^{ 8 }{ -\beta }^{ 8 })) }{ (\alpha +\beta )({ \alpha }^{ 9 }{ -\beta }^{ 9 }) } =\frac { 3({ \alpha }^{ 10 }-{ \beta }^{ 10 }+{ \alpha }^{ 9 }\beta { -\alpha \beta }^{ 9 }) }{ ({ \alpha }^{ 10 }-{ \beta }^{ 10 }+{ \alpha }^{ 9 }\beta { -\alpha \beta }^{ 9 }) } =3$

Now, we can find the remainder of $\left[ \frac { { 3 }^{ 942 } }{ 2014 } \right]$ (Which I guessed using the note.)

Answer: $\boxed{729}$

The fact that a(n)=alpha^n-beta^n implies that a difference equation can be formed.

The following difference equation is formed a(n+2)=6a(n+1)-2a(n).

Clearly a(0)=0.

As alpha=3+sqrt(11) and beta=3-sqrt(11), a(1)=2sqrt(11).

Therefore a(10) and a(8) and a(9) can be generated iteratively.

Giving a(10)=30489792sqrt(11), a(8)=19152sqrt(11), a(9)=4826912sqrt(11).

Therefore, cancelling the sqrt(11)'s, k=(30489792-2 19152)/(2 4826912)=3.

As 3 and 2014 and coprime, Euler's totient theorem can be used. phi(2014)=936.

Therefore k=3^6 mod(11). i.e. The remainder is 3^6=729.

(x – 3)^2 = 11

x = 3 +/- Sqrt (11)

p = 3 + Sqrt (11) and q = 3 - Sqrt (11)

a (n) = [3 + Sqrt (11)]^n – [3 - Sqrt (11)]^n for n >= 1

k = [a (10) – 2 a (8)]/ [2 a (9)] = 3

k^ 942 MOD 2014 = ?

3^942 MOD (2 x 19 x 53) = 729? {Calculator}

With 3^67 = 92709463147897837085761925410587

3^ [14 (67) + 4] MOD 2014

= (92709463147897837085761925410587^14 x 81)/ (2 x 19 x 53)

= (92709463147897837085761925410587 x

92709463147897837085761925410587^13 x 81)/ 2014

= (46032504045629511959166795139+641/ 2014) x 92709463147897837085761925410587^13 x 81

--> (641/ 2014) x 92709463147897837085761925410587^13 x 81

--> (641^14/ 2014) x 81

= (410881^7)/ 2014 x 81

--> (25^7)/ 2014 x 81

= 6103515625/ 2014 X 81

= 494384765625/ 2014

--> 729/ 2014

Remainder = 729

got 3 but stuck to find remainder for a while.