数码和是7倍数的n位数

数论爱好者

northwolves 发表于 2024-11-29 22:17
$a_n=\lfloor \frac{1}{2} \left(1-(-1)^{(n % 2)}\right)+\frac{9\* 10^{n-1}}{11}\rfloor -(-1)^{\lflo ...

在oeis数列网站搜索你的数列8, 82, 818, 8182, 81818, 818182, 8181818, 81818182, 818181818，它给出一个近似答案，
https://oeis.org/A015585，a(n) = 9*a(n-1) + 10*a(n-2)
你的系列和它的系列只是数字不同，对比了一下，规律是相同的，它给出了1000项的值
不懂的是又有a，又有n，算不来

王守恩

数论爱好者 发表于 2024-11-30 01:26
在oeis数列网站搜索你的数列8, 82, 818, 8182, 81818, 818182, 8181818, 81818182, 818181818，它给出一 ...

谢谢 数论爱好者!  我们这串数是这样。
{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, ......
个位数=0,  8,2,8,2,8,2,8,2,8,1,9,  8,2,8,2,8,2,8,2,8,9,1,  8,2,8,2,8,2,8,2,8,1,9,  8,2,8,2,8,2,8,2,8,9,1,  8,2,8,2,8,2,8,2,8,1,9,  8,2,8,2,8,2,8,2,8,9,1,  ......  与下面的不一样。

{1, 9, 91, 909, 9091, 90909, 909091, 9090909, 90909091, 909090909, 9090909091, 90909090909, 909090909091, 9090909090909, 90909090909091, 909090909090909, 9090909090909091, 90909090909090909},
{1, 8, 82, 818, 8182, 81818, 818182, 8181818, 81818182, 818181818, 8181818182, 81818181818, 818181818182, 8181818181818, 81818181818182, 818181818181818, 8181818181818182, 81818181818181818},
{1, 7, 73, 727, 7273, 72727, 727273, 7272727, 72727273, 727272727, 7272727273, 72727272727, 727272727273, 7272727272727, 72727272727273, 727272727272727, 7272727272727273, 72727272727272727},
{1, 6, 64, 636, 6364, 63636, 636364, 6363636, 63636364, 636363636, 6363636364, 63636363636, 636363636364, 6363636363636, 63636363636364, 636363636363636, 6363636363636364, 63636363636363636},
{1, 5, 55, 545, 5455, 54545, 545455, 5454545, 54545455, 545454545, 5454545455, 54545454545, 545454545455, 5454545454545, 54545454545455, 545454545454545, 5454545454545455, 54545454545454545},
{1, 4, 46, 454, 4546, 45454, 454546, 4545454, 45454546, 454545454, 4545454546, 45454545454, 454545454546, 4545454545454, 45454545454546, 454545454545454, 4545454545454546, 45454545454545454},
{1, 3, 37, 363, 3637, 36363, 363637, 3636363, 36363637, 363636363, 3636363637, 36363636363, 363636363637, 3636363636363, 36363636363637, 363636363636363, 3636363636363637, 36363636363636363},
{1, 2, 28, 272, 2728, 27272, 272728, 2727272, 27272728, 272727272, 2727272728, 27272727272, 272727272728, 2727272727272, 27272727272728, 272727272727272, 2727272727272728, 27272727272727272},
{1, 1, 19, 181, 1819, 18181, 181819, 1818181, 18181819, 181818181, 1818181819, 18181818181, 181818181819, 1818181818181, 18181818181819, 181818181818181, 1818181818181819, 18181818181818181},
Table[((11 - a) 10^n + a (-1)^n)/11, {a, 1, 9}, {n, 0, 17}]

{1, 9, 91, 909, 9091, 90909, 909091, 9090909, 90909091, 909090909, 9090909091, 90909090909, 909090909091, 9090909090909, 90909090909091, 909090909090909, 9090909090909091, 90909090909090909},
{1, 8, 82, 818, 8182, 81818, 818182, 8181818, 81818182, 818181818, 8181818182, 81818181818, 818181818182, 8181818181818, 81818181818182, 818181818181818, 8181818181818182, 81818181818181818},
{1, 7, 73, 727, 7273, 72727, 727273, 7272727, 72727273, 727272727, 7272727273, 72727272727, 727272727273, 7272727272727, 72727272727273, 727272727272727, 7272727272727273, 72727272727272727},
{1, 6, 64, 636, 6364, 63636, 636364, 6363636, 63636364, 636363636, 6363636364, 63636363636, 636363636364, 6363636363636, 63636363636364, 636363636363636, 6363636363636364, 63636363636363636},
{1, 5, 55, 545, 5455, 54545, 545455, 5454545, 54545455, 545454545, 5454545455, 54545454545, 545454545455, 5454545454545, 54545454545455, 545454545454545, 5454545454545455, 54545454545454545},
{0, 5, 45, 455, 4545, 45455, 454545, 4545455, 45454545, 454545455, 4545454545, 45454545455, 454545454545, 4545454545455, 45454545454545, 454545454545455, 4545454545454545, 45454545454545455},
{0, 4, 36, 364, 3636, 36364, 363636, 3636364, 36363636, 363636364, 3636363636, 36363636364, 363636363636, 3636363636364, 36363636363636, 363636363636364, 3636363636363636, 36363636363636364},
{0, 3, 27, 273, 2727, 27273, 272727, 2727273, 27272727, 272727273, 2727272727, 27272727273, 272727272727, 2727272727273, 27272727272727, 272727272727273, 2727272727272727, 27272727272727273},
{0, 2, 18, 182, 1818, 18182, 181818, 1818182, 18181818, 181818182, 1818181818, 18181818182, 181818181818, 1818181818182, 18181818181818, 181818181818182, 1818181818181818, 18181818181818182},
Table[Round[((11 - a) 10^n)/11], {a, 1, 9}, {n, 0, 17}]

王守恩

谢谢 mathe！谢谢 northwolves!  收获很大。

数码和是 k 倍数的n位数。公式(1)还是比公式(2)计算速度快一些！
              
公式(1)。Table[NestList[Dot[NestList[RotateRight, IntegerDigits[2^k + 2^9 - 1, 2, k + 1], k], #] &, IntegerDigits[2^k - 2^(k - 9), 2, k + 1], 18][[All, 1]], {k, 9, 38}]

公式(2)。Table[Sum[Coefficient[Sum[x^k, {k, 9}] (Sum[x^k, {k, 9}] + 1)^(k - 1), x, a*r], {r, 9 k/a}], {a, 10, 39}, {k, 25}]

矩阵特征多项式应该是每一串数都有的,  没有找到规律,  写不出来。
         
只有数码和是11倍数的n位数。可以有类似50楼,34楼的通项公式。虽然丑一点。后面的数字串(数码和>11倍数的n位数)就是丑一点, 也没有了。
   
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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}
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},
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},
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},
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},
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},
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},
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},
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},
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},
数码和是30倍数的n位数。{0, 0, 0, 84, 3162, 47631, 453522, 3281421, 22332162, 205725963, 2606606571, 32648395035, 352802138493, 3332321827734, 29612591088978, 273417334179528, 2767675020683610, 29561920127605245},
数码和是31倍数的n位数。{0, 0, 0, 56, 2654, 45002, 464660, 3545628, 23557924, 188937776, 2243299476, 29397578664, 340571458604, 3393236909676, 30527942670240, 271209212091624, 2601695838862040, 27280576744242096},
数码和是32倍数的n位数。{0, 0, 0, 35, 2170, 41712, 468448, 3797615, 25473180, 182582840, 1935097494, 25776043350, 319256668966, 3390440595830, 31607325111220, 277151055094932, 2517848903242216, 25248328413894083},
数码和是33倍数的n位数。{0, 0, 0, 20, 1725, 37917, 464653, 4018401, 27817515, 185848454, 1705890882, 22152625671, 290475344106, 3303078606337, 32425917068544, 288273386247531, 2523491428019804, 23843882173156185},
数码和是34倍数的n位数。{0, 0, 0, 10, 1330, 33787, 453438, 4194465, 30354600, 197124365, 1567012316, 18861860394, 256957531481, 3127926908264, 32648480918010, 300679338989242, 2604345304156432, 23283079235974787},
数码和是35倍数的n位数。{0, 0, 0, 4, 992, 29496, 435303, 4316601, 32887410, 214436852, 1518489653, 16159526133, 221890386874, 2876699973371, 32079491084675, 310549267007240, 2730963678064853, 23577762526023299},
数码和是36倍数的n位数。{0, 0, 0, 1, 714, 25212, 411048, 4379056, 35259774, 235789730, 1551854216, 14202292536, 188352852438, 2571416737970, 30671962664300, 314871844127604, 2867482050433902, 24559929725240762},
数码和是37倍数的n位数。{0, 0, 0, 0, 495, 21087, 381723, 4379055, 37351050, 259374995, 1653501806, 13048390511, 158903196662, 2239291048139, 28507785272627, 311821366378658, 2979627311374646, 25944869574276674},
数码和是38倍数的n位数。{0, 0, 0, 0, 330, 17247, 348558, 4316565, 39069162, 283661048, 1807774694, 12673632929, 135341564137, 1908067516888, 25762940005762, 300826787256636, 3040331338273973, 27404879687078650},
数码和是39倍数的n位数。{0, 0, 0, 0, 210, 13782, 312873, 4194135, 40345140, 307393580, 1999244534, 12995741090, 118641180477, 1602327749724, 22667713720020, 282429021806043, 3032591488352268, 28632735500318184},