数字串的通项公式

王守恩

LinearRecurrence[{3, 3, -11, 3, 3, -1}, {11, 5, 3, 3, 6, 15}, 30]
{11, 5, 3, 3, 6, 15, 43, 121, 351, 1003, 2899, 8322, 24003, 69027, 198903, 572417, 1648739, 4746303, 13668326, 39352691, 113318283, 326274045, 939492531, 2705115363, 7789153783, 22427813890}
{6, 15, 43, 121, 351, 1003, 2899, 8322, 24003, 69027, 198903, 572417, 1648739, 4746303, 13668326, 39352691, 113318283, 326274045, 939492531, 2705115363, 7789153783, 22427813890}
上面这串数, 我是用下面的 "笨办法" 得到的,  这 "笨办法" 应该如何改造？

Module[{g = 1, f = 1, d = 1, c = 1, b = 1, a = 1}, Solve[{x == g, y == g + f, z == f + c, w == g + f + d, s == f + d + c + b, t == d + b + a, k == x + y + z + w + s + t}, {k, t, s, w, z, y, x}]]
{{k -> 15, t -> 3, s -> 4, w -> 3, z -> 2, y -> 2, x -> 1}}

Module[{g = 3, f = 4, d = 3, c = 2, b = 2, a = 1}, Solve[{x == g, y == g + f, z == f + c, w == g + f + d, s == f + d + c + b, t == d + b + a, k == x + y + z + w + s + t}, {k, t, s, w, z, y, x}]]
{{k -> 43, t -> 6, s -> 11, w -> 10, z -> 6, y -> 7, x -> 3}}

Module[{g = 6, f = 11, d = 10, c = 6, b = 7, a = 3}, Solve[{x == g, y == g + f, z == f + c, w == g + f + d, s == f + d + c + b, t == d + b + a, k == x + y + z + w + s + t}, {k, t, s, w, z, y, x}]]
{{k -> 121, t -> 20, s -> 34, w -> 27, z -> 17, y -> 17, x -> 6}}

Module[{g = 20, f = 34, d = 27, c = 17, b = 17, a = 6}, Solve[{x == g, y == g + f, z == f + c, w == g + f + d, s == f + d + c + b, t == d + b + a, k == x + y + z + w + s + t}, {k, t, s, w, z, y, x}]]
{{k -> 351, t -> 50, s -> 95, w -> 81, z -> 51, y -> 54, x -> 20}}

王守恩

1=1——1=1,1×1=1,
2=10——1=1,1×2=2,
3=11——11=3,3×9=9,
4=100——1=1,1×4=4,
5=101——101=5,5×5=25,
6=110——11=3,3×6=18,
7=111——111=7,7×7=49,
8=1000——1=1,1×8=8,
9=1001——1001=9,9×9=81,
10=1010——101=5,5×10=50,
11=1011——1101=13,13×11=143,
12=1100——11=3,3×12=36,
13=1101——1011=11,11×13=143,
14=1110——111=7,7×14=98,
15=1111——1111=15,15×15=225,
16=10000——1=1,1×16=16,
17=10001——10001=17,17×17=289,
18=10010——1001=9,9×18=162,
19=10011——11001=25,25×19=475,
20=10100——101=5,5×20=100,
21=10101——10101=21,21×21=441,
22=10110——1101=13,13×22=286,
23=10111——11101=29,29×23=667,
24=11000——11=3,3×24=72,
25=11001——10011=19,19×25=475,
26=11010——1011=11,11×26=286,
27=11011——11011=27,27×27=729,
28=11100——111=7,7×28=196,
29=11101——10111=23,23×29=667,
30=11110——1111=15,15×30=450,
31=11111——11111=31,31×31=961,
32=100000——1=1,1×32=32,

第2列=第1列的二进制数。
第3列=第2列的颠倒数。
第4列=第3列的十进制数。
第4列×第1列=第7列。
问题来了。
在第7列里找得到4个相同数吗？
直觉: ......应该无解。第7列不会出现4个相同数。

northwolves

王守恩 发表于 2025-3-6 13:39
LinearRecurrence[{3, 3, -11, 3, 3, -1}, {11, 5, 3, 3, 6, 15}, 30]
{11, 5, 3, 3, 6, 15, 43, 121, 351, ...

1. Table[Ceiling[2^n/9 ((8+Sqrt[21] Cos[1/3 ArcTan[37/Sqrt[3]]]+3 Sqrt[7] Sin[1/3 ArcTan[37/Sqrt[3]]])(1/2+Cos[Pi/9])^n+(1-2Cos[2Pi/9])(-Cos[Pi/9])^n +(1-2Cos[4Pi/9])(Cos[2Pi/9])^n )],{n,40}]

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{6,15,43,121,351,1003,2899,8322,24003,69027,198903,572417,1648739,4746303,13668326,39352691,113318283,326274045,939492531,2705115363,7789153783,22427813890,64578615471,185946146107,535411646851,1541654396349,4439020656723,12781643514699,36803288917830,105970822057127,305130867775547,878589229519809,2529797025898535,7284259910134187,20974191063960459,60392775107228994,173894067400571787,500708007398415651,1441731254113828783,4151299681733877817}

northwolves

$\theta=\frac{1}{3} arctan(\frac{37}{\sqrt{3}}\right)$


$a_n= \ceil { \frac{2^n}{9}(  (3 \sqrt{7} \sin \theta+\sqrt{21} \cos \theta+8)(\frac{1}{2}+\cos \frac{\pi }{9})^n+(1-2 \cos \frac{4 \pi }{9}) \cos ^n(\frac{2 \pi }{9})+(1-2 \cos \frac{2 \pi }{9}) (-\cos \frac{\pi }{9})^n))$

northwolves

王守恩 发表于 2025-3-6 19:38
1=1——1=1,1×1=1,
2=10——1=1,1×2=2,
3=11——11=3,3×9=9,

1. x = Table[{n*IntegerReverse[n, 2], n}, {n, 10000}]; y =
   
2. Select[Tally[x[[All, 1]]], #[[2]] > 3 &][[All, 1]]; z =
   
3. Table[{k, Select[x, #[[1]] == k &][[All, 2]]}, {k, y}] // Grid

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8609375        {2375,2755,3125,3625}
10171175        {2527,3059,3325,4025}
30832795        {4555,4835,6377,6769}
17218750        {4750,5510,6250,7250}
35034175        {4775,5539,6325,7337}
38354311        {4847,5371,7141,7913}
39795975        {4959,6099,6525,8025}
20342350        {5054,6118,6650,8050}
41227615        {5819,5995,6877,7085}
51672925        {6575,6775,7627,7859}

northwolves

1. Length@Select[Tally@Table[n*IntegerReverse[n,2],{n,1000000}],#[[2]]>3&]

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1367

northwolves

王守恩 发表于 2025-3-6 13:39
LinearRecurrence[{3, 3, -11, 3, 3, -1}, {11, 5, 3, 3, 6, 15}, 30]
{11, 5, 3, 3, 6, 15, 43, 121, 351, ...

1. f[p_]:=(l=Take[p,-6];x=l[[1]];y=l[[1]]+l[[2]];z=l[[2]]+l[[4]];w=y+l[[3]];s=l[[2]]+l[[3]]+l[[4]]+l[[5]];t=l[[3]]+l[[5]]+l[[6]];k=x+y+z+w+s+t;{k,t,s,w,z,y,x});NestList[f@#&,Array[1&,6],20]

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{{1,1,1,1,1,1},{15,3,4,3,2,2,1},{43,6,11,10,6,7,3},{121,20,34,27,17,17,6},{351,50,95,81,51,54,20},{1003,155,281,226,146,145,50},{2899,421,798,662,427,436,155},{8322,1253,2323,1881,1225,1219,421},{24003,3521,6648,5457,3548,3576,1253},{69027,10286,19229,15626,10196,10169,3521},{198903,29316,55220,45141,29425,29515,10286},{572417,84942,159301,129677,84645,84536,29316},{1648739,243529,458159,373920,243946,244243,84942},{4746303,703105,1320268,1075608,702105,701688,243529},{13668326,2020825,3799669,3098981,2022373,2023373,703105},{39352691,5825459,10944396,8919475,5822042,5820494,2020825},{113318283,16760794,31506407,25689330,16766438,16769855,5825459},{326274045,48284644,90732030,73956531,48272845,48267201,16760794},{939492531,138984526,261228607,212973205,139004875,139016674,48284644},{2705115363,400274523,752223361,613186338,400233482,400213133,138984526},{7789153783,1152383997,2165856314,1765684222,1152456843,1152497884,400274523}}

王守恩

northwolves 发表于 2025-3-6 23:01
8609375        {2375,2755,3125,3625}
10171175        {2527,3059,3325,4025}
30832795        {4555,4835,6377,6769}

Table[Length@Select[Tally@Table[n*IntegerReverse[n, a], {n, 10^6}], #[[2]] > 2 k &], {a, 2, 16}, {k, 5}]

{{1566, 0, 0, 0, 0}, {1371, 1, 0, 0, 0}, {1571, 1, 1, 0, 0}, {1478, 3, 0, 0, 0}, {1697, 0, 0, 0, 0}, {1534, 0, 0, 0, 0}, {1808, 1, 0, 0, 0}, {1506, 4, 0, 0, 0},
{1968, 5, 1, 0, 0}, {2538, 10, 2, 0, 0}, {1794, 3, 0, 0, 0}, {1462, 5, 0, 0, 0}, {1373, 8, 1, 0, 0}, {1905, 10, 2, 0, 0}, {2612, 11, 2, 0, 0}}

没有负担，我只是好奇。浏览一下。

王守恩

3个两两互切的圆, 且相切于同一条直线。3个不同的的半径(r1,r2,r3)都是整数,   譬如：
r1+r2+r3=4+9+36=49,
r1+r2+r3=8+18+72=98,
r1+r2+r3=12+27+108=147,
r1+r2+r3=9+16+144=169,
r1+r2+r3=16+36+144=196,
r1+r2+r3=20+45+180=245,
r1+r2+r3=24+54+216=294,
r1+r2+r3=18+32+288=338,
r1+r2+r3=28+63+252=343,
r1+r2+r3=32+72+288=392,
r1+r2+r3=36+81+324=16+25+400=441,
r1+r2+r3=40+90+360=490,
r1+r2+r3=27+48+432=507,
r1+r2+r3=44+99+396=539,

得到这样一串数——{49, 98, 147, 169, 196, 245, 294, 338, 343, 392, 441, 490, 507, 539, 588, 637, 676, 686, 735, 784, 833, 845, 882, 931, 961, 980, 1014, 1029, 1078, 1127, 1176, 1183, 1225, 1274,
1323, 1352, 1372, 1421, 1470, 1519, 1521, 1568, 1617, 1666, 1690, 1715, 1764, 1813, 1849, 1859, 1862, 1911, 1922, 1960, 2009, 2028, 2058, 2107, 2156, 2197, 2205, 2254, 2303, 2352, 2366, 2401,...}

Union@Flatten@Table[a (1 + b + b^2)^2, {b, 2, 7}, {a, 7^4/(1 + b + b^2)^2}]

northwolves

王守恩 发表于 2025-3-10 10:11
3个两两互切的圆, 且相切于同一条直线。3个不同的的半径(r1,r2,r3)都是整数,   譬如：
r1+r2+r3=4+9+36=49, ...

1. f[n_]:=Union@Flatten@Table[r=(b^2+b+1)^2;r*Range[n/r],{b,2,n^(1/4)}]; f[2500]

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{49,98,147,169,196,245,294,338,343,392,441,490,507,539,588,637,676,686,735,784,833,845,882,931,961,980,1014,1029,1078,1127,1176,1183,1225,1274,1323,1352,1372,1421,1470,1519,1521,1568,1617,1666,1690,1715,1764,1813,1849,1859,1862,1911,1922,1960,2009,2028,2058,2107,2156,2197,2205,2254,2303,2352,2366,2401,2450,2499}