论文标题
广义的斐波那契螺旋
A generalized Fibonacci spiral
论文作者
论文摘要
作为基于复发关系$ f_n = f_ {n-1}+f_ {n-2} $的平面斐波那契螺旋的概括，我们绘制了基于复发性关系的分析解决方案$ g_n = a \ g_ \，g_，g_ {n-1}+b \ b \，g_ n n n n = n n = n n = n n = $ g_0 $和$ g_1 $以及系数$ a $，$ b $，$ c $和$ d $。封闭形式中给出的主坐标对应于交替的偶数或交替的奇数指数的有限总和$ g_ {n} $。对于由直线段（又称螺旋）制成的矩形螺旋，均匀的索引和奇数的定向角点均不位于相互正交的倾斜线上。我们计算交叉点的点，并在伴有螺旋形的情况下显示它们与收敛点一致。在超过螺旋形的情况下，可能会形成一个$ n $依赖的四倍体点。对于拱形螺旋，主坐标之间的插值是通过四分之一椭圆形的弧线进行的。也展示了三维表示。离散序列$ \ {g_n \} $的延续到复杂值函数$ g（t）$，带有真实参数$ t $ t $ t $ t $ t $ t $ r $，在高斯飞机上显示螺旋图和振动曲线，在$ t $ n_0 $的$ t $ for $ g_n $ ass y as zeros中的$ g_n $。此外，我们在变换的Horadam数字方面提供了$ G_N $的矩阵表示，检索应用于$ g_n $的Shannon产品差异身份，并提出了一种替代方法，以查找与$ G_N $相关的各种其他身份和求和。
As a generalization of planar Fibonacci spirals that are based on the recurrence relation $F_n=F_{n-1}+F_{n-2}$, we draw assembled spirals stemming from analytic solutions of the recurrence relation $G_n=a\, G_{n-1}+b\, G_{n-2}+c\, d\,^n$, with positive real initial values $G_0$ and $G_1$ and coefficients $a$, $b$, $c$, and $d$. The principal coordinates given in closed-form correspond to finite sums of alternating even- or alternating odd-indexed terms $G_{n}$. For rectangular spirals made of straight line segments (a.k.a. spirangles), the even-indexed and the odd-indexed directional corner points asymptotically lie on mutually orthogonal oblique lines. We calculate the points of intersection and show them in the case of inwinding spirals to coincide with the point of convergence. In the case of outwinding spirals, an $n$-dependent quadruple of points of intersection may form. For arched spirals, interpolation between principal coordinates is performed by means of arcs of quarter-ellipses. A three-dimensional representation is exhibited, too. The continuation of the discrete sequence $\{G_n\}$ to the complex-valued function $G(t)$ with real argument $t$$\in$$R$, exhibiting spiral graphs and oscillating curves in the Gaussian plane, subsumes the values $G_n$ for $t$$\in$$N_0$ as the zeros. Besides, we provide a matrix representation of $G_n$ in terms of transformed Horadam numbers, retrieve the Shannon product difference identity as applied to $G_n$, and suggest a substitution method for finding a variety of other identities and summations related to $G_n$.
