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The series

$1$

$x+1$

$x^2+x+1$

$x^3+x^2+x+1$

$x^4+x^3+x^2+x+1$

$x^5+x^4+x^3+x^2+x+1$

$x^6+x^5+x^4+x^3+x^2+x+1$

$x^7+x^6+x^5+x^4+x^3+x^2+x+1$

$x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1$

$x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1$

$.........$

can be written as the exponential function $f(x)$ $=$ $(x^n-1)/(x-1)$ for any $x$ value.

A similar sequence, defining the same fields as $f(x)$, is there an exponential function $g(x)$ form for all degrees, or exponents $n$?

$1$

$x+6$

$x^2+13x+43$

$x^3+18x^2+112x+240$

$x^4+23x^3+204x^2+832x+1321$

$x^5+28x^4+319x^3+1860x^2+5589x+6966$

$x^6+33x^5+459x^4+3457x^3+14947x^2+35409x+36163$

$x^7+38x^6+624x^5+5752x^4+32244x^3+110408x^2+215104x+185280$

$x^8+43x^7+814x^6+8872x^5+61006x^4+271720x^3+768460x^2+1268416x+942001$

$x^9+48x^8+1029x^7+12942x^6+105366x^5+576766x^4+2127568x^3+5116564x^2+7313257x+4763526$

$.........$

Thanks if one can show how to represent this sequence as an exponential function. This is namely the minimum polynomial of $x^2+2x+5$ $\pmod {(x^n-1)/(x-1)}$ for all $n$.

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    An exponential function is $a^x$ . Perhaps you mean polynomial quotient. – 2017-02-24
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    what is this mod thing exactly ? – 2017-02-24
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    @zwim : e.g. https://en.wikipedia.org/wiki/Modulo_operation – 2017-02-24
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    @ user90369 this wasn't the question. – 2017-02-24
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    I think he means $P(y) = \prod (y - (\zeta^2 + 2\zeta + 5))$ where $\zeta$ varies among all the roots of $1+x+\ldots+x^n$ but it doesn't work ... – 2017-02-24

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