Sequence

This is a quite complicate solution, but I think it can be traced back to @Anirudh Sreekumar solution (the answer is the same)

$\begin{aligned} \color{#D61F06} 6 & \\ \color{#3D99F6} 14 \color{#333333} & = \color{#D61F06} 6 \color{#333333} *3 - \color{#20A900} 4 \\ \color{#D61F06} 36 \color{#333333} & = \color{#3D99F6} 14 \color{#333333} *3 - \color{#20A900} 6 \\ \color{#3D99F6} 98 \color{#333333} & = \color{#D61F06} 36 \color{#333333} *3 - \color{#20A900} 10 \\ \color{#D61F06} 276 \color{#333333} & = \color{#3D99F6} 98 \color{#333333} *3 - \color{#20A900} 18 \\ \color{#3D99F6} \boxed{794} \color{#333333} & = \color{#D61F06} 276 \color{#333333} *3 - \color{#20A900} 34 \end{aligned}$

$\color{#333333}$ The regularity between the green numbers:

$\begin{aligned} -6 & = -4-2^1 \\ -10 & = -6-2^2 \\ -18 & = -10-2^3 \\ -34 & = -18-2^4 \end{aligned}$

The given sequence is $1^n+2^n+3^n$

The next number occurs at $n=6$ and is equal to $1^6+2^6+3^6=794$