数字串的通项公式

northwolves

王守恩 发表于 2024-9-29 19:23
A167430——有通项公式吗？ 谢谢！

1, 2, 1, 3, 2, 4, 1, 3, 5, 2, 4, 6, 1, 3, 5, 7, 2, 4, 6, 8, 1, 3,  ...

或者

$a_k=2 k - \floor[\sqrt[k] + \frac{1}{2}]^2 - (\floor(\sqrt[k] )+ 1/2 - round[\sqrt[k]] + \floor[\sqrt[k - 1]])^2$

王守恩

northwolves 发表于 2024-9-30 23:04
$a_k=2 \left(k-\lfloor \sqrt{k}+\frac{1}{2}\rfloor ^2\right)+(round(\sqrt{k})-\lfloor \sqrt{k-1}\r ...

1. Table[2 (k - Round[Sqrt[k]]^2) + (Round[Sqrt[k]] - Floor[Sqrt[k - 1]]) (2 Round[Sqrt[k]] - 1), {k, 42}]

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{1, 2, 1, 3, 2, 4, 1, 3, 5, 2, 4, 6, 1, 3, 5, 7, 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8, 10, 1, 3, 5, 7, 9, 11, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 13, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 15, 2, 4, 6, 8, 10, 12, 14, 16}

1. Table[k - Round[Sqrt[k]] Floor[Sqrt[k - 1]], {k, 72}]

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{1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8}

northwolves

王守恩 发表于 2024-10-1 09:40
{1, 2, 1, 3, 2, 4, 1, 3, 5, 2, 4, 6, 1, 3, 5, 7, 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8, 10, 1, 3, ...

1. Table[k-Floor[Sqrt[k]-1/2]*Floor[Sqrt[k]+1/2],{k,72}]

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{1,2,1,2,3,4,1,2,3,4,5,6,1,2,3,4,5,6,7,8,1,2,3,4,5,6,7,8,9,10,1,2,3,4,5,6,7,8,9,10,11,12,1,2,3,4,5,6,7,8,9,10,11,12,13,14,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}

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  0,    1,    3,   6,  10,  15,  21, 28, 36,
45,  46,  48,  51,  55,  60,  66, 73, 81,
90,  91,  93,  96,100,105,111,118,126,
135,136,138,141,145,150,156,163,171,
180,181,183,186,190,195,201,208,216,
225,226, 228,231,235,240,246,253,261,
270,271,273,276,280,285,291,298,306,
......
{1, 3, 6, 10, 15, 21, 28, 36, 45, 46, 48, 51, 55, 60, 66, 73, 81, 90, 91, 93, 96, 100, 105, 111, 118, 126, 135, 136, 138, 141, 145, 150, 156, 163, 171, 180, 181, 183, 186, 190, 195, 201, 208, 216, 225, 226, 228, 231,
235, 240, 246, 253, 261, 270, 271, 273, 276, 280, 285, 291, 298, 306, 315, 316, 318, 321, 325, 330, 336, 343, 351, 360, 361, 363, 366, 370, 375, 381, 388, 396, 405, 406, 408, 411, 415, 420, 426, 433, 441, 450}
Table[(Mod[n, 9] (Mod[n, 9] - 9) + 10 n)/2, {n, 60}]

王守恩

兼得。Table[(A - Mod[n, A]) Mod[n, A], {A, 15}, {n, A}]
{ 0},
{ 1,  0},
{ 2,  2,  0},
{ 3,  4,  3,  0},
{ 4,  6,  6,  4,  0},
{ 5,  8,  9,  8,   5,  0},
{ 6, 10, 12, 12, 10,  6, 0},
{ 7, 12, 15, 16, 15, 12,  7, 0},
{ 8, 14, 18, 20, 20, 18, 14,  8,  0},
{ 9, 16, 21, 24, 25, 24, 21, 16,  9,  0},
{10, 18, 24, 28, 30, 30, 28, 24, 18, 10,  0},
{11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 0},
{12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12, 0},
{13, 24, 33, 40, 45, 48, 49, 48, 45, 40, 33, 24, 13, 0},
{14, 26, 36, 44, 50, 54, 56, 56, 54, 50, 44, 36, 26, 14, 0}

northwolves

王守恩 发表于 2018-12-4 13:49
有这样一串数：1, 2, 5, 11, 22, 45, 91, 182, 365, 731, 1462, 2925, 5851, 11702, 23405, 46811,
9362 ...

$a_n=\frac{5}{7} \left(2^n-1\right)+\frac{1}{21} \left(3 \sqrt{3} \sin \left(\frac{2 \pi  n}{3}\right)-\cos \left(\frac{2 \pi  n}{3}\right)+1\right)$

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northwolves 发表于 2024-10-5 23:43
$a_n=\frac{5}{7} \left(2^n-1\right)+\frac{1}{21} \left(3 \sqrt{3} \sin \left(\frac{2 \pi  n}{3}\ri ...

有这样一串数——详见9楼。
{1, 2, 5, 11, 22, 45, 91, 182, 365, 731, 1462, 2925, 5851, 11702, 23405, 46811, 93622, 187245, 374491, 748982, 1497965, 2995931, 5991862, 11983725, 23967451, 47934902, 95869805, 191739611, 383479222, 766958445}
公式(1)。Table[FromDigits[PadRight[{}, n, {1, 0, 1}], 2], {n, 40}]
{1, 2, 5, 11, 22, 45, 91, 182, 365, 731, 1462, 2925, 5851, 11702, 23405, 46811, 93622, 187245, 374491, 748982, 1497965, 2995931, 5991862, 11983725, 23967451, 47934902, 95869805, 191739611, 383479222, 766958445}
公式(2)。Table[Floor[(5*2^n)/7], {n, 30}]
{1, 2, 5, 11, 22, 45, 91, 182, 365, 731, 1462, 2925, 5851, 11702, 23405, 46811, 93622, 187245, 374491, 748982, 1497965, 2995931, 5991862, 11983725, 23967451, 47934902, 95869805, 191739611, 383479222, 766958445}
相近的一串数：OEIS——没有这串数了。
{1, 3, 6, 12, 25, 50, 100, 201, 402, 804, 1609, 3218, 6436, 12873, 25746, 51492, 102985, 205970, 411940, 823881, 1647762, 3295524, 6591049, 13182098, 26364196, 52728393, 105456786, 210913572, 421827145, 843654290}
公式(1)不知道怎么用。公式(2)可以这样用。
Table[Floor[(11*2^n)/7], {n, 0, 60}]
{1, 3, 6, 12, 25, 50, 100, 201, 402, 804, 1609, 3218, 6436, 12873, 25746, 51492, 102985, 205970, 411940, 823881, 1647762, 3295524, 6591049, 13182098, 26364196, 52728393, 105456786, 210913572, 421827145, 843654290}

northwolves

王守恩 发表于 2024-10-7 19:34
有这样一串数——详见9楼。
{1, 2, 5, 11, 22, 45, 91, 182, 365, 731, 1462, 2925, 5851, 11702, 23405,  ...

$a_n=\frac{11*\ 2^{n-1}}{7}-\frac{1}{21} \left(3 \sqrt{3} \sin \left(\frac{2 \pi  n}{3}\right)-\cos \left(\frac{2 \pi  n}{3}\right)+7\right)$

northwolves

or $a_n=\frac{1}{14} \left(11*\ 2^n-2 (2^{n+1} \mod 7)\right)$

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{0, 0, 1, 3, 23, 177, 1553, 14963, 157931, 1814453, 22566237, 302267423, 4340478951, 66541218865, 1084982173641, 18752743351339, 342523093859011, 6593167693927885, 133408305489947029,
2831112931136162775, 62878579846490149375, 1458746608689369440265, 35287049763090967922369, 888576906273358995242787, 23257202390087081114367899, 631813050547326757658171621,
17791673379141486111275196749, 518691169689842352465788751887, 15637460560236108808905870638487, 486992031871106518198669000829537, 15650827841213007202590962838172473, ......}

1. RecurrenceTable[{a[n + 7] == a[n + 6] (n + 10) - a[n + 5] (2 n + 7) - a[n + 4] (3 n + 22) + a[n + 3] (4 n + 9) + (3 a[n + 2] - a[n]) (n + 2) - a[n + 1] (2 n + 9),
   
2. a[1] == a[2] == 0, a[3] == 1, a[4] == 3, a[5] == 23, a[6] == 177, a[7] == 1553}, a, {n, 29}]

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