数字串的通项公式

王守恩

northwolves 发表于 2023-10-4 18:04
{1,2,3,5,7,10,13,18,23,30,37,47,57,70,83,101,119,142,165,195,225,262,299,346,393,450,507,577,647 ...

谢谢 northwolves！您有方法(我有题目)! 这算式可以调吗？谢谢！
第2串:a(1)=1, a(n) = a(n-1) + floor((n+0)/2))

1. Table[If[n<2,n,a[n-1]]+Floor[(n+0)/2],{n,36}]

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第3串:a(1)=1, a(n) = a(n-1) + floor((n+1)/3))

1. Table[If[n<2,n,a[n-1]]+Floor[(n+1)/3],{n,36}]

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第4串:a(1)=1, a(n) = a(n-1) + floor((n+2)/4))

1. Table[If[n<2,n,a[n-1]]+Floor[(n+2)/4],{n,36}]

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第5串:a(1)=1, a(n) = a(n-1) + floor((n+3)/5))

1. Table[If[n<2,n,a[n-1]]+Floor[(n+3)/5],{n,36}]

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第6串:a(1)=1, a(n) = a(n-1) + floor((n+4)/6))

1. Table[If[n<2,n,a[n-1]]+Floor[(n+4)/6],{n,36}]

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第7串:a(1)=1, a(n) = a(n-1) + floor((n+5)/7))

1. Table[If[n<2,n,a[n-1]]+Floor[(n+5)/7],{n,36}]

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第2串:1, 2, 3, 5, 7, 10, 13, 17, 21, 26, 31, 37, 43, 50, 57, 65, 73, 82, 91, 101, 111, 122, 133, 145, 157, 170, 183, 197, 211, 226, 241,  
第3串:1, 2, 3, 4, 6, 8, 10, 13, 16, 19, 23, 27, 31, 36, 41, 46, 52, 58, 64, 71, 78, 85, 93, 101, 109, 118, 127, 136, 146, 156, 166, 177, 188,  
第4串:1, 2, 3, 4, 5, 7, 9, 11, 13, 16, 19, 22, 25, 29, 33, 37, 41, 46, 51, 56, 61, 67, 73, 79, 85, 92, 99, 106, 113, 121, 129, 137, 145, 154, 163,   
第5串:1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 19, 22, 25, 28, 31, 35, 39, 43, 47, 51, 56, 61, 66, 71, 76, 82, 88, 94, 100, 106, 113, 120, 127, 134, 141,
第6串:1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 22, 25, 28, 31, 34, 37, 41, 45, 49, 53,57, 61, 66, 71, 76, 81, 86, 91, 97, 103, 109, 115, 121, 127, 134,  
第7串:1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 25, 28, 31, 34, 37, 40, 43, 47, 51, 55, 59, 63, 67, 71, 76, 81, 86, 91, 96, 101, 106, 112, 118, 124,
第8串:1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 28, 31, 34, 37, 40, 43, 46, 49, 53, 57, 61, 65, 69, 73, 77, 81, 86, 91, 96, 101, 106, 111, 116,
第9串:1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 96, 101, 106, 111,
  OEIS 没有这样连贯的。

王守恩

二进制数，且要求出现000,111的数，譬如：
a(1)=0,
a(2)=0,
a(3)=01, 111,
a(4)=03, 1000,1110,1111,
a(5)=08, 10000,10001,10111,11000,11100,11101,11110,11111,
a(6)=19, 100000,100001,100010,100011,100111,101000,101110,101111,110000,110001,110111,111000,111001,111010,111011,111100,111101,111110,111111,
a(7)=43,
a(8)=91,
......
0,0,1,3,8,18,40,91,......
这是怎样的一串数？

northwolves

1. a[n_]:=Length@Select[IntegerDigits[Range[2^(n-1),2^n-1],2],SequenceCount[#,{0,0,0}]+SequenceCount[#,{1,1,1}]>0&];Table[a[n],{n,12}]

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{0, 0, 1, 3, 8, 19, 43, 94, 201, 423, 880, 1815}

northwolves

1. Table[Round[2^(r - 1) - ((Sqrt[5] + 1)/2)^(r + 1)/Sqrt[5]], {r, 30}]

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{0,0,1,3,8,19,43,94,201,423,880,1815,3719,7582,15397,31171,62952,126891,255379,513342,1030865,2068495,4147936,8313583,16655823,33358014,66791053,133703499,267603416,535524643}

northwolves

A008466                a(n) = 2^n - Fibonacci(n+2).

0, 0, 1, 3, 8, 19, 43, 94, 201, 423, 880, 1815, 3719, 7582, 15397, 31171, 62952, 126891, 255379, 513342, 1030865, 2068495, 4147936, 8313583, 16655823, 33358014, 66791053, 133703499, 267603416, 535524643, 1071563515, 2143959070, 4289264409, 8580707127

northwolves

1. Table[Length@Select[Join@@Permutations/@IntegerPartitions[n],Max@@#>2&],{n,15}]

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1. LinearRecurrence[{3, -1, -2}, {0, 0, 1}, 40]

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王守恩

northwolves 发表于 2023-10-7 12:31
A008466                a(n) = 2^n - Fibonacci(n+2).

0, 0, 1, 3, 8, 19, 43, 94, 201, 423, 880, 1815, 3719, 7582, 1 ...

好歹也是一个。0, 0, 1, 3, 8, 19, 43, 94, 201, 423, 880, 1815, 3719, 7582, 15397, 31171, 62952, 126891, 255379, 513342, 1030865, 2068495, 4147936, 8313583, 16655823, 33358014, 66791053, 133703499, 267603416 ......

1. RecurrenceTable[{a[n]==2^(n-4)+2^(n-3)+2a[n-2]+a[n-3],a[1]==a[2]==0,a[3]==1},a[n],{n,1,9}]

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说明:
a[n]=2^(n-4)+a[n-3]+2a[n-2]+a[n-2]+2^(n-3)
a(01)=0,
a(02)=0,
a(03)=001,
a(04)=003=001+00+000+000+002,
a(05)=008=002+00+001+001+004,
a(06)=019=004+01+003+003+008,
a(07)=043=008+03+008+008+016,
a(08)=094=016+08+019+019+032,
a(09)=201=032+19+043+043+064,
a(10)=423=064+43+094+094+128,
a(11)=880=128+94+201+201+256,
......
a(n)=****=(1)+(2)+(3) +(4) +(5),
说明：
(1):头部=1000, 尾部=全排列,
(2):头部=1001,后面的"1"=a(n-3)头部的"1",
(3):头部=101, 后面的"1"=a(n-2)头部的"1",
(4):头部=110,后面的"0"=a(n-2)头部添"0",
(5):头部=111, 尾部=全排列。

王守恩

5, 7, 11, 17, 19, 29, 31, 41, 61, 67, 71, 79, 89, 97, 101, 107, 109, 127, 131, 137, 139, 151, 157, 167, 197, 211, 227, 229, 239, 269, 277, 307, 317, 331, 347, 349, 379, 401, 409, 419, 431, 439, 449, 461, ...
A136052。这样也可以。       

1. Select[Prime@Range@90,PrimeQ[7#-6]&]

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王守恩

将 2,3,4,5,6,…,n 排成数列{ak:k=1,2,3,4,5,…,n-1},使 ak 都是 k 的倍数,有几种排法?
a(00)=0,
a(01)=0,
a(02)=1,{2},
a(03)=1,{3,2},
a(04)=2,{2,4,3},{4,2,3},
a(05)=1,{5,2,3,4},
a(06)=3,{2,6,3,4,5}{3,2,6,4,5}{6,2,3,4,5},
a(07)=1,{7,2,3,4,5,6},
a(08)=3,{2,8,3,4,5,6,7},{4,2,3,8,5,6,7},{8,2,3,4,5,6,7},
a(09)=2,{3,2,9,4,5,6,7,8},{9,2,3,4,5,6,7,8},
a(10)=3,
a(11)=1,
a(12)=5,
a(13)=1,
a(14)=3,
a(15)=3,
a(16)=4,
......
{0, 0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 7, 2, 3, 3, 5, 1, 7, 1, 5, 3, 3, 3, 8, 1, 3, 3, 7, 1, 7, 1, 5, 5, 3, 1, 9, 2, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 11, 1, 3, 5, 6, 3, 7, 1, 5, ...

1. Prepend[DivisorSigma[0, Range[99]] - 1, 0]

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A032741        给出了上面的通项公式。
若改: 将 3,4,5,6,…,n 排成数列{ak:k=1,2,3,4,5,…,n-2},使 ak 都是 k 的倍数,有几种排法?
a(00)=0,
a(01)=0,
a(02)=0,
a(03)=1,{3},
a(04)=1,{3,4},
a(05)=1,{5,4,3},
a(06)=1,{5,6,3,4},
a(07)=1,{7,6,3,4,5},
a(08)=2,{7,4,3,8,5,6},{7,8,3,4,5,6},
a(09)=4,{3,4,8,9,5,6,7},{3,8,9,4,5,6,7},{9,4,3,8,5,6,7}{9,8,3,4,5,6,7},
a(10)=2,{3,10,9,4,5,6,7,8},{9,10,3,4,5,6,7,8},
a(11)=1,{11,10,3,4,5,6,7,8,9},
a(12)=3,{11,4,3,12,5,6,7,8,9,10},{11,6,3,4,5,12,7,8,9,10},{11,12,3,4,5,6,7,8,9,10},
a(13)=3,{13,4,3,12,5,6,7,8,9,10,11},{13,6,3,4,5,12,7,8,9,10,11},{13,12,3,4,5,6,7,8,9,10,11},
a(14)=1,{13,14,3,4,5,6,7,8,9,10,11,12},
a(15)=3,
......
还可以有通项公式吗？谢谢各位！

王守恩

409#前半部分有问题。
将2,3,4,5,6,…,n排成数列{a(k):k=1,2,3,4,5,…,n-1},使a(k)都是k的倍数,有几种排法?
a(00)=1,{1},   把409#的表示方法也简化一下。
a(01)=1,{2},   {2}表示:a(1)=2,
a(02)=1,{3},   {3}表示:a(1)=3,
a(03)=2,{2,4},{4},  {2,4}表示:a(1)=2,a(2)=4,{4}表示:a(1)=4,
a(04)=1,{5},
a(05)=3,{2,6}{3,6}{6},{2,6}表示:a(1)=2,a(2)=6,{3,6}表示:a(1)=3,a(3)=6,
a(06)=1,{7},
a(07)=4,{2,4,8},{2,8},{4,8},{8},   {2,4,8}表示:a(1)=2,a(2)=4,a(4)=8,......
a(08)=2,{3,9},{9},
a(09)=3,{2,10},{5,10},{10},
a(10)=1,{11},
a(11)=8,{2,4,12},{2,6,12},{2,12},{3,6,12},{3,12},{4,12},{6,12},{12},
a(12)=1,{13},
a(13)=3,{2,14},{7,14},{14},
a(14)=3,{3,15},{5,15},{15},
a(15)=8,{2,4,8,16},{2,4,16},{2,8,16},{2,16},{4,8,16},{4,16},{8,16},{16},
a(16)=1,{17},
a(17)=8,{2,6,18},{2,18},{3,6,18},{3,9,18},{3,18},{6,18},{9,18},{18},
a(18)=1,{19},
a(19)=8,{2,4,20},{2,10,20},{2,20},{4,20},{5,10,20},{5,20},{10,20,}{20},
{1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3, 8, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 4, 8, 1, 13, 1, 16, 3, 3, 3, 26, 1, 3, 3, 20, 1, 13, 1, 8, 8, 3, 1, 48, 2,
8, 3, 8, 1, 20, 3, 20, 3, 3, 1, 44, 1, 3, 8, 32, 3, 13, 1, 8, 3, 13, 1, 76, 1, 3, 8, 8, 3, 13, 1, 48, 8, 3, 1, 44, 3, 3, 3, 20, 1, 44, 3, 8, 3, 3, 3, 112}

1. a[0]=1;a[1]=1;a[n_]:=a[n]=a/[url=home.php?mod=space&uid=6175]@[/url]Most[Divisors[n]]//Total;a/@Range[96]

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