数列的通项公式

northwolves

$y(1)=2,y(n)=\frac{\left(10 n^2-19 n-2\right) y(n-1)+1728 n^2-5024 n+3504}{10 n^2-59 n+94}$

前30项：
{2, 23, 671, 3671, 11903, 29399, 61343, 114071, 195071, 312983, 477599, 699863, 991871, 1366871, 1839263, 2424599, 3139583, 4002071, 5031071, 6246743, 7670399, 9324503, 11232671, 13419671, 15911423, 18734999, 21918623, 25491671, 29484671, 33929303}

nyy

这么复杂，还是算了吧

王守恩

RecurrenceTable[{y[1] == 2, (45 - 39 n + 10 n^2) y[1 + n] == 208 - 1568 n + 1728 n^2 - (11 - n - 10 n^2) y[n]}, y, {n, 40}]——质数是不可以有公式的。

{2, 23, 671, 3671, 11903, 29399, 61343, 114071, 195071, 312983, 477599, 699863, 991871, 1366871, 1839263, 2424599, 3139583, 4002071, 5031071, 6246743, 7670399, 9324503, 11232671,
13419671, 15911423, 18734999, 21918623, 25491671, 29484671, 33929303, 38858399, 44305943, 50307071, 56898071, 64116383, 72000599, 80590463, 89926871, 100051871, 111008663}

mathe

通项不重要吧，不过这个数列为什么看上去都是整数？有什么背景吗？

northwolves

mathe 发表于 2025-11-4 16:21
通项不重要吧，不过这个数列为什么看上去都是整数？有什么背景吗？

$x$ 取这个数列，$768 n^4 + 48 n^2 x + x^2 $ 是一个平方数。

1. Table[{n, Values@FindInstance[y^2 == 768 n^4 + 48 n^2 x + x^2, {x, y},  PositiveIntegers]}, {n, 0,30}]

复制代码

{{0, {{2, 2}}},{1,{{23,49}}},{2,{{671,769}}},{3,{{3671,3889}}},{4,{{11903,12289}}},{5,{{29399,30001}}},{6,{{61343,62209}}},{7,{{114071,115249}}},{8,{{195071,196609}}},{9,{{312983,314929}}},{10,{{477599,480001}}},{11,{{699863,702769}}},{12,{{991871,995329}}},{13,{{1366871,1370929}}},{14,{{1839263,1843969}}},{15,{{2424599,2430001}}},{16,{{3139583,3145729}}},{17,{{4002071,4009009}}},{18,{{5031071,5038849}}},{19,{{6246743,6255409}}},{20,{{7670399,7680001}}},{21,{{9324503,9335089}}},{22,{{11232671,11244289}}},{23,{{13419671,13432369}}},{24,{{15911423,15925249}}},{25,{{18734999,18750001}}},{26,{{21918623,21934849}}},{27,{{25491671,25509169}}},{28,{{29484671,29503489}}},{29,{{33929303,33949489}}},{30,{{38858399,38880001}}}}

northwolves

5楼的数据并不是最小的x:

1. Table[{n, Values@Solve[y^2 == 768 n^4 + 48 n^2 x + x^2 && x < 5000, {x, y}, PositiveIntegers][[1]]}, {n, 0, 20}]

复制代码

{{0,{1,1}},{1,{23,49}},{2,{26,134}},{3,{11,259}},{4,{104,536}},{5,{110,790}},{6,{44,1036}},{7,{112,1456}},{8,{416,2144}},{9,{99,2331}},{10,{440,3160}},{11,{913,4169}},{12,{176,4144}},{13,{26,4706}},{14,{329,5719}},{15,{195,6405}},{16,{1664,8576}},{17,{2482,10234}},{18,{396,9324}},{19,{4598,14174}},{20,{1760,12640}}}

wayne

1. ans = {2, 23, 671, 3671, 11903, 29399, 61343, 114071, 195071, 312983,
   
2.    477599, 699863, 991871, 1366871, 1839263, 2424599, 3139583,
   
3.    4002071, 5031071, 6246743, 7670399, 9324503, 11232671, 13419671,
   
4.    15911423, 18734999, 21918623, 25491671, 29484671, 33929303};
   
5. expr = FindGeneratingFunction[ans, x];
   
6. {expr, SeriesCoefficient[expr, {x, 0, n}]}

复制代码

先得到生成函数$g(x)=\frac{3 x^5-38 x^4-526 x^3-576 x^2-13 x-2}{(x-1)^5}$，然后得到通项公式的新表达
\[a(n)=\begin{array}{cc}
\{ &
\begin{array}{cc}
48 n^4-24 n^2-1 & n>0 \\
2 & n=0 \\
\end{array}
\\
\end{array}\]

northwolves

$x_n=48*n^4 - 192*n^3 + 264*n^2 - 144*n + 23,   y_n=48 n^4+1 (n>1)$

northwolves

$y^2=768 n^4 + 48 n^2 x + x^2 $ 的通解应该是有的吧？