数列的通项公式
$y(1)=2,y(n)=\frac{\left(10 n^2-19 n-2\right) y(n-1)+1728 n^2-5024 n+3504}{10 n^2-59 n+94}$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} 这么复杂，还是算了吧 RecurrenceTable[{y == 2, (45 - 39 n + 10 n^2) y == 208 - 1568 n + 1728 n^2 - (11 - n - 10 n^2) y}, 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} 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？ 本帖最后由 northwolves 于 2025-11-4 16:53 编辑
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。
Table[{n, Values@FindInstance}, {n, 0,30}]
{{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}}}} 5楼的数据并不是最小的x:
Table[{n, Values@Solve[]}, {n, 0, 20}]
{{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}}} ans = {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};
expr = FindGeneratingFunction;
{expr, SeriesCoefficient}
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
\[a(n)=\begin{array}{cc}
\{ &
\begin{array}{cc}
48 (n-1)^4-24 (n-1)^2-1 & n>1 \\
2 & n=1 \\
\end{array}
\\
\end{array}\] 本帖最后由 northwolves 于 2025-11-4 17:07 编辑
$x_n=48*n^4 - 192*n^3 + 264*n^2 - 144*n + 23, y_n=48 n^4+1 (n>1)$ $y^2=768 n^4 + 48 n^2 x + x^2 $ 的通解应该是有的吧？ Constant differences
The differences of order 4 in the difference table of depth 1 appear to become constant.
The next few terms would be 9335089, 11244289, 13432369, 15925249.
2 49 76938891228930001622091152491966093149294800017027699953291370929184396924300013145729400900950388496255409768000193350891124428913432369
47 72031208400177123220853040 81360118320165072222768292560375600 473040 586032 715728 86328010298401216560142459216550881909200 2188080
673240052809312144962083228320 36960 46752 57696 69792 83040 97440 112992 129696 147552 166560 186720 208032 230496 254112 278880
1727288040325184 6336 7488 8640 9792 10944 12096 13248 14400 15552 16704 17856 19008 20160 21312 22464 23616 24768
1153115211521152 1152 1152 1152 1152 1152 1152 1152 1152 1152 1152 1152 1152 1152 1152 1152 1152
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