求证：当 n=1,2,3,… 时，a(n)=∑(k=1,2n+1){cos[kπ／(n+1)]}^2=n

王守恩

这是怎么来的？  n=1, 2, 3, 4, 5, 6, 7, 8, 9, ......

\(\displaystyle a(n)=\sum_{k=1}^{2n+1}\cos^2\big(\frac{k\ \pi}{n+1}\big)=1,2,3,4,5,...\)

luyuanhong

王守恩

我就好奇：这么多无理数加起来刚好是1,2,3,4,5,6,7,8,9,...。
{{0., 1., 0.},
{0.25, 0.25, 1., 0.25, 0.25},
{0.5, 0., 0.5, 1., 0.5, 0., 0.5},
{0.654508, 0.0954915, 0.0954915, 0.654508, 1., 0.654508, 0.0954915, 0.0954915, 0.654508},
{0.75, 0.25, 0., 0.25, 0.75, 1., 0.75, 0.25, 0., 0.25, 0.75},
{0.811745, 0.38874, 0.0495156, 0.0495156, 0.38874, 0.811745, 1., 0.811745,
   0.38874, 0.0495156, 0.0495156, 0.38874, 0.811745},
{0.853553, 0.5, 0.146447, 0., 0.146447, 0.5, 0.853553,
   1., 0.853553, 0.5, 0.146447, 0., 0.146447, 0.5, 0.853553},
{0.883022, 0.586824, 0.25, 0.0301537, 0.0301537, 0.25, 0.586824, 0.883022,
   1., 0.883022, 0.586824, 0.25, 0.0301537, 0.0301537, 0.25, 0.586824, 0.883022},
{0.904508, 0.654508, 0.345492, 0.0954915, 0., 0.0954915, 0.345492, 0.654508, 0.904508,
   1.,  0.904508, 0.654508, 0.345492, 0.0954915, 0., 0.0954915, 0.345492, 0.654508, 0.904508},
{0.920627, 0.707708, 0.428843, 0.17257, 0.0202535, 0.0202535, 0.17257, 0.428843,
   0.707708, 0.920627, 1., 0.920627, 0.707708, 0.428843, 0.17257, 0.0202535, 0.0202535,
   0.17257, 0.428843, 0.707708, 0.920627},
{0.933013, 0.75, 0.5, 0.25, 0.0669873, 0., 0.0669873, 0.25, 0.5, 0.75, 0.933013, 1., 0.933013,
   0.75, 0.5, 0.25, 0.0669873, 0., 0.0669873, 0.25, 0.5, 0.75, 0.933013},
{0.942728, 0.784032, 0.560268, 0.322698, 0.125745, 0.0145291, 0.0145291, 0.125745,
   0.322698,0.560268, 0.784032, 0.942728, 1., 0.942728, 0.784032, 0.560268, 0.322698,
   0.125745, 0.0145291,0.0145291, 0.125745, 0.322698, 0.560268, 0.784032,  0.942728},
{0.950484, 0.811745, 0.61126, 0.38874, 0.188255, 0.0495156, 0., 0.0495156, 0.188255,
   0.38874, 0.61126, 0.811745,  0.950484, 1., 0.950484, 0.811745, 0.61126, 0.38874,
   0.188255,0.0495156,0., 0.0495156, 0.188255, 0.38874, 0.61126, 0.811745, 0.950484},
{0.956773, 0.834565, 0.654508, 0.447736, 0.25, 0.0954915,0.0109262, 0.0109262, 0.0954915,
   0.25, 0.447736, 0.654508, 0.834565, 0.956773, 1., 0.956773, 0.834565, 0.654508, 0.447736,
   0.25,0.0954915,0.0109262,0.0109262,0.0954915,0.25,0.447736,0.654508,0.834565,0.956773},
{0.96194, 0.853553, 0.691342, 0.5, 0.308658, 0.146447, 0.0380602, 0., 0.0380602, 0.146447,
   0.308658, 0.5, 0.691342, 0.853553, 0.96194, 1., 0.96194, 0.853553, 0.691342, 0.5, 0.308658,
   0.146447, 0.0380602, 0., 0.0380602, 0.146447, 0.308658, 0.5, 0.691342, 0.853553, 0.96194},
{0.966236, 0.869504,  0.722869, 0.546134, 0.363169, 0.198683, 0.0748914, 0.00851345,
   0.00851345, 0.0748914, 0.198683, 0.363169, 0.546134, 0.722869, 0.869504, 0.966236,
   1., 0.966236, 0.869504, 0.722869, 0.546134, 0.363169, 0.198683, 0.0748914, 0.00851345,
   0.00851345, 0.0748914, 0.198683, 0.363169, 0.546134, 0.722869, 0.869504, 0.966236},
{0.969846, 0.883022, 0.75, 0.586824, 0.413176, 0.25, 0.116978, 0.0301537, 0., 0.0301537,
   0.116978, 0.25, 0.413176, 0.586824, 0.75, 0.883022, 0.969846, 1., 0.969846, 0.883022, 0.75,
   0.586824, 0.413176, 0.25, 0.116978, 0.0301537, 0., 0.0301537, 0.116978, 0.25, 0.413176,
   0.586824, 0.75, 0.883022, 0.969846}}