数字串的通项公式

northwolves · 发表于 2024-10-30 18:53:19

$a_n=10^{n-1}-8\* 9^{n-2},a_1=1$

王守恩 · 发表于 2024-11-1 13:50:13

{\[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi], \[Pi]}

Table[(Gamma[n + 1/2]/((2 n - 1)!! 2^-n))^2, {n, 29}]

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王守恩 · 发表于 2024-11-4 15:38:00

{1, 4, 10, 20, 35, 56, 084, 120, 165, 220, 286, 364, 455, 560, 0680, 0816, 0969, 1140, 1330},
{1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470},
{1, 6, 18, 40, 75, 126, 196, 288, 405, 550, 726, 936, 1183, 1470, 1800, 2176, 2601, 3078, 3610},
{1, 7, 22, 50, 95, 161, 252, 372, 525, 715, 946, 1222, 1547, 1925, 2360, 2856, 3417, 4047, 4750},
{1, 8, 26, 60, 115, 196, 308, 456, 645, 880, 1166, 1508, 1911, 2380, 2920, 3536, 4233, 5016, 5890},
{1, 9, 30, 70, 135, 231, 364, 540, 765, 1045, 1386, 1794, 2275, 2835, 3480, 4216, 5049, 5985, 7030},
{1, 10, 34, 80, 155, 266, 420, 624, 885, 1210, 1606, 2080, 2639, 3290, 4040, 4896, 5865, 6954, 8170},
{1, 11, 38, 90, 175, 301, 476, 708, 1005, 1375, 1826, 2366, 3003, 3745, 4600, 5576, 6681, 7923, 9310},
{1, 12, 42, 100, 195, 336, 532, 792, 1125, 1540, 2046, 2652, 3367, 4200, 5160, 6256, 7497, 8892, 10450}

Table[((k (n - 1) + 3) n (n + 1))/6, {k, 9}, {n, 19}]

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王守恩 · 发表于 2024-11-5 11:25:04

n位数, 各位数字之和等于n。

这样的1位数有1个。

这样的2位数有2个。

这样的3位数有6个。

这样的4位数有20个。

这样的5位数有70个。

这样的6位数有252个。

这样的7位数有924个。

这样的8位数有3432个。

这样的9位数有12870个。

northwolves · 发表于 2024-11-7 17:27:21

已确认。
A000984
Central binomial coefficients: binomial(2*n,n) = (2*n)!/(n!)^2.

1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520, 601080390, 2333606220, 9075135300, 35345263800, 137846528820, 538257874440, 2104098963720, 8233430727600, 32247603683100, 126410606437752, 495918532948104, 1946939425648112

王老师可以尝试用生成函数证明或解释一下

王守恩 · 发表于 2024-11-14 07:40:30

a(1)=1,{1},
a(2)=2,{12},{21},
a(3)=3,{112},{121},{211},
a(4)=6,{1122},{1212},{1221},{2112},{2121},{2211},6=3*2
a(5)=10,{11122},{11212},{11221},{12112},{12121},{12211},{21112},{211121},{21211},{22111},10=6+4?
a(6)=20,{111222},{112122},{112212},{112221},{121122},{121212},{121221},{122112},{122121},{122211},20=10*2
a(7)=35,{2111122},{2111212},{2111221},{2112112},{2112121},{2112211},{2121112},{2121121},{2121211},{2122111},{2211112},{2211121},{2211211},{2212111},{2221111},35=20+15?
a(8)=70,70=35*2
a(9)=126,126=70+56?
a(10)=252,252=126*2
a(11)=462,462=252+210?
a(12)=924,924=462*2
a(13)=1716,1716=924+792?
a(14)=3432,3432=1716*2
a(15)=6435,6435=3432+3003?
a(16)=12870,12870=6435*2
......
1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462, 924, 1716, 3432, 6435, 12870, 24310, 48620, 92378, 184756, 352716, 705432,

是这串数吗？要不就是？处有问题了。

如果是这串数, OEIS应该有上面的条文清晰简单些。

王守恩 · 发表于 2024-11-14 07:47:03

这通项公式还是比OEIS——A131708——好一些。

{1, 2, 3, 5, 10, 21, 43, 86, 171, 341, 682, 1365, 2731, 5462, 10923, 21845, 43690, 87381, 174763, 349526, 699051, 1398101, 2796202, 5592405, 11184811}

Table[Round[(2^n + Sin[n*Pi/3])/3], {n, 25}]

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王守恩 · 发表于 2024-11-15 15:04:44

楼上的通项公式可以有更好的。

(2^n + 2 Cos[(n - 2)*Pi/3])/3=(2^n - 2 Cos[(n + 1)*Pi/3])/3=Round[(2^n + Sin[n*Pi/3])/3]

{1, 2, 3, 5, 10, 21, 43, 86, 171, 341, 682, 1365, 2731, 5462, 10923, 21845, 43690, 87381, 174763, 349526, 699051, 1398101, 2796202, 5592405, 11184811}

王守恩 · 发表于 2024-11-22 16:26:21

数码和是11倍数的n位数。

这样的1位数有0个。
这样的2位数有8个。
这样的3位数有82个。
这样的4位数有818个。

0,
8,
82,
818,
8182,
81818,
818182,
8181818,
81818182,
818181818,
8181818181,
81818181819,
818181818182,
8181818181818,
81818181818182,
818181818181818,
8181818181818182,
81818181818181818,
818181818181818182,
8181818181818181818,
81818181818181818182,
818181818181818181819,
8181818181818181818181,
81818181818181818181818,
818181818181818181818182,
8181818181818181818181818,
81818181818181818181818182,
818181818181818181818181818,
8181818181818181818181818182,
81818181818181818181818181818,
818181818181818181818181818182,
8181818181818181818181818181818,
81818181818181818181818181818181,
818181818181818181818181818181819,
8181818181818181818181818181818182,
81818181818181818181818181818181818,
818181818181818181818181818181818182,
8181818181818181818181818181818181818,
81818181818181818181818181818181818182,
818181818181818181818181818181818181818,
8181818181818181818181818181818181818182,
81818181818181818181818181818181818181818,
818181818181818181818181818181818181818182,
8181818181818181818181818181818181818181819,
81818181818181818181818181818181818181818181,
818181818181818181818181818181818181818181818,
8181818181818181818181818181818181818181818182,
81818181818181818181818181818181818181818181818,
818181818181818181818181818181818181818181818182,
8181818181818181818181818181818181818181818181818,
81818181818181818181818181818181818181818181818182,
818181818181818181818181818181818181818181818181818,
8181818181818181818181818181818181818181818181818182,
81818181818181818181818181818181818181818181818181818,
818181818181818181818181818181818181818181818181818181,
8181818181818181818181818181818181818181818181818181819,
81818181818181818181818181818181818181818181818181818182,
818181818181818181818181818181818181818181818181818181818,
8181818181818181818181818181818181818181818181818181818182,
81818181818181818181818181818181818181818181818181818181818,
818181818181818181818181818181818181818181818181818181818182,
8181818181818181818181818181818181818181818181818181818181818,
81818181818181818181818181818181818181818181818181818181818182,
818181818181818181818181818181818181818181818181818181818181818,
8181818181818181818181818181818181818181818181818181818181818182,
81818181818181818181818181818181818181818181818181818181818181819,
818181818181818181818181818181818181818181818181818181818181818181}

Table[(9*10^n - Cos[n Pi/11] - Cos[3 n Pi/11] - Cos[5 n Pi/11] - Cos[7 n Pi/11] - Cos[9 n Pi/11] - Cos[13 n Pi/11] - Cos[15 n Pi/11] - Cos[17 n Pi/11]
- Cos[19 n Pi/11] - Cos[21 n Pi/11] + Cos[(n + 1) Pi/11] + Cos[3 (n + 1) Pi/11] + Cos[5 (n + 1) Pi/11] - Cos[(13 n + 2) Pi/11] - Cos[(15 n + 4) Pi/11]
- Sin[(14 n + 3) Pi/22] + Sin[3 (14 n + 3) Pi/22] - Sin[(18 n + 7) Pi/22] + Sin[(34 n + 1) Pi/22] + Sin[(38 n + 5) Pi/22])/11, {n, 0, 22}] // FullSimplify

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王守恩 · 发表于 2024-11-29 15:18:36

谢谢 mathe！！！这公式还是厉害些！
Table[NestList[Dot[NestList[RotateRight, IntegerDigits[2^k + 2^9 - 1, 2, k + 1], k], #] &, IntegerDigits[2^k - 2^(k - 9), 2, k + 1], 17][[All, 1]], {k, 9, 28}]

(01)。数码和是1倍数的n位数。{9, 90, 900, 9000, 90000, 900000, 9000000, 90000000, 900000000, 9000000000, 90000000000, 900000000000, 9000000000000, 90000000000000, 900000000000000, 9000000000000000, 90000000000000000}
(02)。数码和是2倍数的n位数。{4, 45, 450, 4500, 45000, 450000, 4500000, 45000000, 450000000, 4500000000, 45000000000, 450000000000, 4500000000000, 45000000000000, 450000000000000, 4500000000000000, 45000000000000000}
(03)。数码和是3倍数的n位数。{3, 30, 300, 3000, 30000, 300000, 3000000, 30000000, 300000000, 3000000000, 30000000000, 300000000000, 3000000000000, 30000000000000, 300000000000000, 3000000000000000, 30000000000000000}
(04)。数码和是4倍数的n位数。{2, 22, 224, 2249, 22500, 225002, 2250004, 22500004, 225000000, 2249999992, 22499999984, 224999999984, 2250000000000, 22500000000032, 225000000000064, 2250000000000064, 22500000000000000}
(05)。数码和是5倍数的n位数。{1, 18, 180, 1800, 18000, 180000, 1800000, 18000000, 180000000, 1800000000, 18000000000, 180000000000, 1800000000000, 18000000000000, 180000000000000, 1800000000000000, 18000000000000000}
(06)。数码和是6倍数的n位数。{1, 14, 151, 1503, 14997, 149991, 1500009, 15000027, 149999973, 1499999919, 15000000081, 150000000243, 1499999999757, 14999999999271, 150000000000729, 1500000000002187, 14999999999997813}
(07)。数码和是7倍数的n位数。{1, 12, 126, 1282, 12860, 128598, 1285774, 12857176, 128571220, 1285713534, 12857141804, 128571429416, 1285714293398, 12857142874408, 128571428581010, 1285714285653962, 12857142856925458}
(08)。数码和是8倍数的n位数。{1, 11, 112, 1124, 11248, 112496, 1124992, 11249985, 112499976, 1124999972, 11249999992, 112500000074, 1125000000280, 11250000000692, 112500000001384, 1125000000002340, 11250000000003264}
(09)。数码和是9倍数的n位数。{1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 10000000000000000}
(10)。数码和是10倍数的n位数。{0, 9, 90, 900, 9000, 90000, 900000, 9000000, 90000000, 900000000, 9000000000, 90000000000, 900000000000, 9000000000000, 90000000000000, 900000000000000, 9000000000000000, 90000000000000000}
(11)。数码和是11倍数的n位数。{0, 8, 82, 818, 8182, 81818, 818182, 8181818, 81818182, 818181818, 8181818181, 81818181819, 818181818182, 8181818181818, 81818181818182, 818181818181818, 8181818181818182, 81818181818181818}
(12)。数码和是12倍数的n位数。{0, 7, 76, 748, 7504, 74993, 749994, 7500059, 74999910, 750000001, 7500000200, 74999999501, 750000000944, 7499999999031, 74999999998090, 750000000009683, 7499999999984066, 75000000000005245}
(13)。数码和是13倍数的n位数。{0, 6, 72, 684, 6933, 69297, 692049, 6923265, 69231861, 692302884, 6923085159, 69230782122, 692307584595, 6923077148598, 69230769343458, 692307690281355, 6923076928701546, 69230769228514479}
(14)。数码和是14倍数的n位数。{0, 5, 70, 630, 6375, 64601, 642645, 6425599, 64297713, 642856767, 6428421494, 64286170240, 642857602231, 6428564424863, 64285730047092, 642857181352496, 6428571116489480, 64285714761152326}
(15)。数码和是15倍数的n位数。{0, 4, 69, 603, 5817, 60378, 602133, 5987904, 59994303, 600225342, 5999520723, 59997327243, 600015197949, 6000007240428, 59999717638617, 600000601182243, 6000003351236328, 59999980942125519}
(16)。数码和是16倍数的n位数。{0, 3, 66, 599, 5332, 55956, 569584, 5618444, 56103129, 562918205, 5627520744, 56235963162, 562469405642, 5625371327946, 56250000000692, 562491672466081, 5625015387280082, 56250158324512242}
(17)。数码和是17倍数的n位数。{0, 2, 61, 607, 5005, 51090, 539103, 5335482, 52635691, 528573446, 5303297495, 52955278635, 529146620809, 5293987371496, 52948567080626, 529408195240026, 5293918597046695, 52941466543240964}
(18)。数码和是18倍数的n位数。{0, 1, 54, 616, 4884, 46300, 503700, 5118916, 49881084, 496175516, 5003824484, 50123007764, 499876992236, 4996043649836, 50003956350164, 500127249814052, 4999872750185948, 49995907208582972}
(19)。数码和是19倍数的n位数。{0, 0, 45, 615, 4950, 42459, 461055, 4904064, 47998149, 468183583, 4708257444, 47541493289, 474905200991, 4731718185444, 47318535154702, 473822169622558, 4738808901078066, 47365330213246505}
(20)。数码和是20倍数的n位数。{0, 0, 36, 597, 5124, 40242, 415140, 4624980, 46674240, 448239640, 4428061744, 44928061744, 452795788288, 4508531241376, 44902816850752, 449457017475904, 4502885274370048, 45027958693108096}
(21)。数码和是21倍数的n位数。{0, 0, 28, 564, 5318, 39814, 373327, 4257781, 45247819, 436940959, 4203612028, 42237989270, 430403180040, 4318885704619, 42873915907996, 427136419253877, 4280690450082956, 42906423299276119},
(22)。数码和是22倍数的n位数。{0, 0, 21, 519, 5465, 40909, 342531, 3830055, 43107021, 430324609, 4067803414, 39791380024, 405828327164, 4132191361796, 41228700104788, 408346458179408, 4072530775092692, 40873078728369612},
(23)。数码和是23倍数的n位数。{0, 0, 15, 465, 5520, 42999, 326796, 3402165, 40005165, 421929825, 4015305614, 38039545603, 380763807239, 3918682413337, 39702546659487, 393767982767586, 3895329112873457, 38917753783501426},
(24)。数码和是24倍数的n位数。{0, 0, 10, 405, 5460, 45464, 326556, 3039256, 36126849, 406147125, 4001610250, 37219365121, 359631428364, 3678322478042, 37935664292588, 381392881265011, 3762350098218829, 37219852374909939},
(25)。数码和是25倍数的n位数。{0, 0, 6, 342, 5283, 47757, 339291, 2789685, 31952940, 380512875, 3965587680, 37181174265, 346581662075, 3445279720300, 35772868790700, 367441289159325, 3661162712706600, 35966744400571500},
(26)。数码和是26倍数的n位数。{0, 0, 3, 279, 4998, 49488, 360864, 2675682, 28056429, 346151442, 3856245834, 37461129369, 342920665075, 3266992419028, 33407288613937, 349150070540774, 3555835163480783, 35114955454707434},
(27)。数码和是27倍数的n位数。{0, 0, 1, 219, 4620, 50413, 386727, 2694120, 24930511, 306816399, 3649461057, 37492291378, 346450438830, 3176913779355, 31275582443221, 326843073213153, 3410727440409699, 34351481452917616},
(28)。数码和是28倍数的n位数。{0, 0, 0, 165, 4170, 50412, 412764, 2823018, 22890318, 267392223, 3351046590, 36813730077, 352560385658, 3178406721221, 29819823777614, 303860739629215, 3214285558244740, 33280426266832651},
(29)。数码和是29倍数的n位数。{0, 0, 0, 120, 3675, 49467, 435765, 3030213, 22046520, 232528065, 2989841118, 35193107787, 356061516838, 3245017470529, 29280382725973, 284803132371826, 2986317296390089, 31665783163960501},

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[原创] 数字串的通项公式

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