数字串的通项公式

northwolves

1. RSolve[{a[1] == 1, a[n] == a[n - 1]*2^n - n!}, a[n], n]
   
2. 
   
3. RSolve[{b[1] == 1, b[n] == b[n - 1]*2^n - n}, b[n], n]

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$\{a(n)\to 2^{\frac{n^2}{2}+\frac{n}{2}-1} \left(\sum _{K[1]=0}^{n-1} -2^{-\frac{1}{2} K[1] (K[1]+3)} (K[1]+1)!+2\right)\}$

$\{b(n)\to 2^{\frac{n^2}{2}+\frac{n}{2}-1} \left(\sum _{K[1]=-1}^{n-1} -2^{-\frac{1}{2} K[1] (K[1]+3)} (K[1]+1)+2\right)\}$

northwolves

$b(n)\to 2^{\frac{n^2}{2}+\frac{n}{2}-1} \left(\sum _{k=-2}^{n-1} -2^{-\frac{1}{2}k^2-\frac{3k}{2}-1} \left(k^2+3 k+2\right)+2\right)$

northwolves

$\frac{a_n}{b_n}=\frac{2-\sum _{k=0}^{n-1} 2^{-\frac{1}{2} k (k+3)} (k+1)!}{2-\sum _{k=-2}^{n-1}2^{-\frac{1}{2} k^2-\frac{3 k}{2}-1} \left(k^2+3k+2\right)}$

1        1.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
2        2.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
3        5.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
4        6.181818181818181818181818181818181818181818181818181818181818181818181818181818181818181818181818182
5        6.142235123367198838896952104499274310595065312046444121915820029027576197387518142235123367198838897
6        6.128825865002836074872376630743051616562677254679523539421440726035167328417470221213840045377197958
7        6.127962915300912582521325616335305124174609488277380843495394546059148053060388132953013805371977881
8        6.127935150298689888341984388055590635077852956201404850602377691175325391846480595906838342114421629
9        6.127934659929888948655680092552788232325595296561105642900553603223233774279322590210486098366979738
10        6.127934655137934431594879458646852123959734808395378664366502707548733668042408722960977549002825187
11        6.127934655112194264719775435249581026495480300145001963490559785724874545988492137357582845906355431
12        6.127934655112118853385802931683324087632432236740299478346728770582294908735278956766910518958866794
13        6.127934655112118733714387879726615850208366579219366828194639042278242406234970664462027749893152775
14        6.127934655112118733612129582094949168192620626575522577565224772636452840777146135890463248078327107
15        6.127934655112118733612082771960826443268099203260457751416869220207718771864942753251159319465960259
16        6.127934655112118733612082760532571036240903532026878786741341748350385413175173194154085788857183107
17        6.127934655112118733612082760531088794862678672319873766242066445294967558748658465326832191107694150
18        6.127934655112118733612082760531088693085240286877384191542162320184035945495022521376484772158512373
19        6.127934655112118733612082760531088693081551910776226611560221810997946329963814796809981289459952594
20        6.127934655112118733612082760531088693081551840426035473147491904320640230162807159316401994766509023
21        6.127934655112118733612082760531088693081551840425331015857840668707511662524367004234067135735477709
22        6.127934655112118733612082760531088693081551840425331012162815135907031798201445971730044651466224124
23        6.127934655112118733612082760531088693081551840425331012162805004835553143931632542573656738060470674
24        6.127934655112118733612082760531088693081551840425331012162805004821060529931329841843244339690084507
25        6.127934655112118733612082760531088693081551840425331012162805004821060519133490974907051878994073471
26        6.127934655112118733612082760531088693081551840425331012162805004821060519133486791498276842346098352
27        6.127934655112118733612082760531088693081551840425331012162805004821060519133486791497435284091394551
28        6.127934655112118733612082760531088693081551840425331012162805004821060519133486791497435284003613185
29        6.127934655112118733612082760531088693081551840425331012162805004821060519133486791497435284003613180
30        6.127934655112118733612082760531088693081551840425331012162805004821060519133486791497435284003613180

王守恩

northwolves 发表于 2025-5-21 18:22
$\frac{a_n}{b_n}=\frac{2-\sum _{k=0}^{n-1} 2^{-\frac{1}{2} k (k+3)} (k+1)!}{2-\sum _{k=-2}^{n-1}2^{- ...

我就想让6.1279346551121187336120与下面的数扯上关系。

$\frac{2}{1}+\frac{6}{2}+\frac{12}{6}+\frac{20}{24}+\frac{30}{120}+\frac{42}{720}+\frac{56}{5040}\cdots+\frac{n(n+1)}{n!}=3e$

王守恩

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64}

\(\D\bigg[\sum_{k=1}^{2n^2}\big(\sqrt{2 k} - \sqrt{2 k - 1}\ \ \big)\bigg]=n\) ——答案没问题——如何证明？

王守恩

{3, 11, 95, 1414, 31619, 980328, 39966975, 2063473712, 131165658459, 10041515879680, 909567637557215, 96070344004816128, 11688399779985830355, 1621144844290431509504, 254042974238965752088575,
44630552378618565828990976, 8730289639752813502468309547, 1890024460764917200545258602496, 450405494714907780225378366952479, 117579766849364197687078369206272000, 33477582935288039868454992643985264163}

Table[Sum[(n k)^(n + 1)/(n + 1)^(k + 1), {k, Infty}], {n, 20}]

王守恩

northwolves 发表于 2025-5-21 18:16
$b(n)\to 2^{\frac{n^2}{2}+\frac{n}{2}-1} \left(\sum _{k=-2}^{n-1} -2^{-\frac{1}{2}k^2-\frac{3k}{2}-1 ...

A094500——Least number k such that (n+1)^k / n^k >= 2.
1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, ——通项公式怎么调？谢谢！

northwolves

王守恩 发表于 2025-5-29 13:19
A094500——Least number k such that (n+1)^k / n^k >= 2.
1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11 ...

1. a=Ceiling@Log[1 + 1/Range@100, 2]

复制代码

王守恩

northwolves 发表于 2025-5-29 23:46

a=Ceiling@Log[1 + 1/Range@100, 2]——是这样一串数。

{1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 44, 45, 45, 46}

$a(1)=1,(\frac{2}{1})^0<2≤(\frac{2}{1})^1$

$a(2)=2,(\frac{3}{2})^1<2<(\frac{3}{2})^2$

$a(3)=3,(\frac{4}{3})^2<2<(\frac{4}{3})^3$

$a(4)=4,(\frac{5}{4})^3<2<(\frac{5}{4})^4$

$a(5)=4,(\frac{6}{5})^3<2<(\frac{6}{5})^4$

$a(6)=5,(\frac{7}{6})^4<2<(\frac{7}{6})^5$

$a(7)=6,(\frac{8}{7})^5<2<(\frac{8}{7})^6$

$a(8)=6,(\frac{9}{8})^5<2<(\frac{9}{8})^6$

王守恩

$a(1)=1,(\frac{2}{1})^1<\pi<(\frac{2}{1})^2$

$a(2)=2,(\frac{3}{2})^2<\pi<(\frac{3}{2})^3$

$a(3)=3,(\frac{4}{3})^3<\pi<(\frac{4}{3})^4$

$a(4)=5,(\frac{5}{4})^5<\pi<(\frac{5}{4})^6$

$a(5)=6,(\frac{6}{5})^6<\pi<(\frac{6}{5})^7$

$a(6)=7,(\frac{7}{6})^7<\pi<(\frac{7}{6})^8$

$a(7)=8,(\frac{8}{7})^8<\pi<(\frac{8}{7})^9$

$a(8)=9,(\frac{9}{8})^9<\pi<(\frac{9}{8})^{10}$

{1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74}

a = Floor@Log[1 + 1/Range@65, Pi]——OEIS有一串相近的数——A029927——nautical miles to statute miles——嗨！我们的比她有意思多了！！！