This could not be more imaginary

Algebra Level 5

Let f(n)=a1n+a2n+a3nf(n) = { a }_{ 1 }^{ n }+{ a }_{ 2 }^{ n }+{ a }_{ 3 }^{ n }.

Given that

f(1)=a1+a2+a3=0f(2)=a12+a22+a32=if(3)=a13+a23+a33=ii{ f(1)=a }_{ 1 }+{ a }_{ 2 }+{ a }_{ 3 }=0\\ f(2)={ a }_{ 1 }^{ 2 }+{ a }_{ 2 }^{ 2 }+{ a }_{ 3 }^{ 2 }=i\\ f(3)={ a }_{ 1 }^{ 3 }+{ a }_{ 2 }^{ 3 }+{ a }_{ 3 }^{ 3 }={ i }^{ i }

and that (f(4)+f(7))=a+b(eπc)d-(f(4)+f(7))=\frac { a+b\left( { e }^{ \frac { \pi }{ c } } \right) }{ d }

Find a+b+c+da+b+c+d given that they are integers .... Assumptions\textbf{Assumptions}

\bullet i=1i=\sqrt { -1 }

\bullet (e)(e) is Euler's number or (e=limn(1+1n)n) \left(e=\lim _{ n\rightarrow \infty }{ { \left( 1+\frac { 1 }{ n } \right) }^{ n } }\right)

....

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