777. Lissajous Curves

For coprime positive integers a and b, let Ca,b be the curve defined by:

x=cos(at)y=cos(b(tπ10))

where t varies between 0 and 2π.

For example, the images below show C2,5 (left) and C7,4 (right):

Define d(a,b)=(x2+y2), where the sum is over all points (x, y) at which Ca,b crosses itself.

For example, in the case of C2,5 illustrated above, the curve crosses itself at two points: (0.31, 0) and (-0.81, 0), rounding coordinates to two decimal places, yielding d(2,5)=0.75. Some other examples are d(2,3)=4.5, d(7,4)=39.5, d(7,5)=52, and d(10,7)=23.25.

Let s(m)=d(a,b), where this sum is over all pairs of coprime integers a,b with 2am and 2bm. You are given that s(10)=1602.5 and s(100)=24256505.

Find s(106). Give your answer in scientific notation rounded to 10 significant digits; for example s(100) would be given as 2.425650500e7.

777. 利萨茹曲线

对于互质的两个正整数 ab,定义曲线 Ca,b 为满足下述参数方程的曲线,其中 t 取遍 [0,2π]

x=cos(at)y=cos(b(tπ10))

例如,下图左、右两侧分别展示了曲线 C2,5C7,4

d(a,b) 为:Ca,b 中所有自相交的点 (x,y) 到原点距离的平方的和。即 d(a,b)=(x2+y2)

观察左图,C2,5 有两个自相交点:(0.31,0)(0.81,0)(所有坐标四舍五入到小数点后两位),从而 d(2,5)=0.75。已知:d(2,3)=4.5d(7,4)=39.5d(7,5)=52d(10,7)=23.25

s(m)=2a,bma,bd(a,b)。已知 s(10)=1602.5s(100)=24256505

s(106)。将你的答案用科学记数法表示,并把小数部分四舍五入至第 10 位有效数字。如 s(100) 应被表示为 2.425650500e7。


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