因式分解 x^(3a-1)+x+1 ，a=1,2,3,4,5,6,……

王守恩

\(因式分解x^{3a-1}+x+1,    a=1,2,3,4,5,6,...\)

Treenewbee

1. Table[Factor[x^(3 a - 1) + x + 1], {a, 6}]

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{1 + x + x^2, (1 + x + x^2) (1 - x^2 + x^3), (1 + x + x^2) (1 - x^2 +
    x^3 - x^5 + x^6), (1 + x + x^2) (1 - x^2 + x^3 - x^5 + x^6 - x^8 +
     x^9), (1 + x + x^2) (1 - x^2 + x^3 - x^5 + x^6 - x^8 + x^9 -
    x^11 + x^12), (1 + x + x^2) (1 - x^2 + x^3 - x^5 + x^6 - x^8 +
    x^9 - x^11 + x^12 - x^14 + x^15)}

时空伴随者

\(x^{3a-1}+x+1=(x^2+x+1)*(x^{3a-3}-x^{3a-4}+\dots+x^3-x^2+1)\),    a=2,3,4,5,6,...

王守恩

0, -2, 3, -5, 6, -8, 9, -11, 12, -14, 15, -17, 18, -20, 21, ......

连我都被这通项公式卡住了。

时空伴随者

王守恩 发表于 2023-5-29 08:30
0, -2, 3, -5, 6, -8, 9, -11, 12, -14, 15, -17, 18, -20, 21, ......

连我都被这通项公式卡住了。

\(a_n=(-1)^n(3\lfloor\frac{n+1}2\rfloor-n\ mod\ 2)\), n=0,1,2,...

Treenewbee

\[a_n=\frac{(-1)^n*(6 n + 1)-1}{4}\]

王守恩

谢谢Treenewbee！谢谢时空伴随者！受两位启发，可以有

\(a_{n}=\frac{3n\cos(n\pi)-\mod(n, 2)}{2}\)
0, -2, 3, -5, 6, -8, 9, -11, 12, -14, 15, -17, 18, -20, 21, ......

\(a_{n}=2 n - \lfloor n/2\rfloor\cos(n\pi)\)
0, -2, 3, -5, 6, -8, 9, -11, 12, -14, 15, -17, 18, -20, 21, ......

LinearRecurrence[{1, 1, -1}, {0, 2, 3}, 40]
0, 2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, ......