倒过来写的平方数

王守恩

乘胜追击！！！接楼上。(1) = 楼上第 1 行 = 从 0 到 n,  只能 + 1, × 2,  最快几步？(2) = 楼上第 2 行 = 从 0 到 n,  只能 + 1, × 3,  最快几步？
(1): {1, 2, 3, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 5, 6, 6, 7, 6, 7, 7, 8, 6, 7, 7, 8, 7, 8, 8, 9, 6, 7, 7, 8, 7, 8, 8, 9, 7, 8, 8, 9, 8, 9, 9, 10, 7, 8, 8, 9, 8, 09, 9, 10, 8, 9, 9, 10, 9, 10, 10, 11, 7, 8, 8, 9, 8, 9, 9, 10, 8, 9, 9, 10, 9, 10, 10, 11, 08, 9, 9,10, 9, 10},
(2): {1, 2, 2, 3, 4, 3, 4, 5, 3, 4, 5, 4, 5, 6, 5, 6, 7, 4, 5, 6, 5, 6, 7, 6, 7, 8, 4, 5, 6, 5, 6, 7, 6, 7, 8, 5, 6, 7, 6, 7, 8, 7, 8, 9, 6, 7, 08, 7, 8, 9, 8, 9, 10, 5, 06, 7, 6, 7, 08, 7, 08, 09, 06, 7, 8, 7, 8, 9, 8, 9, 10, 7, 8, 9, 08, 9, 10, 09, 10, 11, 5, 6, 07, 6, 07},
(3): {1, 2, 2, 3, 4, 3, 4, 4, 3, 4, 5, 4, 5, 5, 5, 5, 6, 4, 5, 5, 5, 6, 7, 5, 6, 6, 4, 5, 6, 5, 6, 6, 6, 7, 8, 5, 6, 6, 6, 6, 7, 6, 7, 7, 6, 7, 08, 6, 7, 7, 7, 7, 08, 5, 06, 6, 6, 7, 08, 6, 07, 07, 06, 7, 8, 7, 8, 8, 8, 9, 10, 6, 7, 7, 07, 7, 08, 07, 08, 07, 5, 6, 07, 6, 07},

(1)=从 0 到 n,  只能 + 1, × 2,  最快几步？
(2)=从 0 到 n,  只能 + 1, × 3,  最快几步？
(3)=从 0 到 n,  只能 + 1, × 2, × 3,  最快几步？

a(1)=1,0+1,
a(2)=2,0+1×2,
a(3)=2,0+1×3,
a(4)=3,0+1×3+1,
a(5)=4,0+1×3+1+1,
a(6)=3,0+1×3×2,
a(7)=4,0+1×3×2+1,
a(8)=4,0+1×2×2×2,
a(9)=3,0+1×3×3,
a(10)=4,0+1×3×3+1,
a(11)=5,0+1×3×3+1+1,
a(12)=4,a(6)×2,
a(13)=5,a(12)+1,
a(14)=5,a(7)×2,
a(15)=5,a(5)×3,
a(16)=5,a(8)×2,
a(17)=6,a(16)+1,
a(18)=4,a(6)×3,
a(19)=5,a(18)+1,
a(20)=5,a(10)×2,
a(21)=5,a(7)+1,
a(22)=6,a(11)×2,
a(23)=7,a(23)+1,
a(24)=5,a(8)×3,
a(25)=6,a(24)+1,
a(26)=6,a(13)×2,
a(27)=4,a(9)×3,
a(28)=5,a(27)+1,
a(29)=6,a(28)+1,
a(30)=5,a(10)×3,
a(31)=6,a(30)+1,
a(32)=6,a(16)×2,

得到这样一串数——1, 2, 2, 3, 4, 3, 4, 4, 3, 4, 5, 4, 5, 5, 5, 5, 6, 4, 5, 5, 5, 6, 7, 5, 6, 6, 4, 5, 6, 5, 6, 6, 6, 7, 8, 5, 6, 6, 6, 6, 7, 6, 7, 7, 6, 7, 8, 6, 7, 7, 7, 7, 8, 5, 6, 6, 6, 7, 8, 6, 7, 7, 6, 7, 8, 7, 8, 8, 8, 9, 10, 6, 7, 7,

A056796——有这串数——就是没有通项公式。

northwolves

A056796——有这串数——就是没有通项公式。
--------------------------------------------------------------------

1. f[0] = 0; f[n_] := f[n] = (r = f[n - 1]; s = r;
   
2.    If[Mod[n, 2] == 0, s = Min[f[n/2], r]];
   
3.    If[Mod[n, 3] == 0, s = Min[f[n/3], r], r]; 1 + s);
   
4. Array[f, 1000]

复制代码

王守恩

有这么一串数——{1, 4, 9, 18, 46, 52, 61, 63, 94, 121, 144, 148, 163, 169, 423, 441, 484, 487, 522, 526, 652, 675, 676, 691, 925, 927, 961, 982, ......}

规律是这样: 把平方数倒过来写；然后按从小到大重新排列。

问2个问题。

(1), 第2025个数是几？——6148035。——没有规律只能靠电脑。
a = Union@Table[IntegerReverse[k^2], {k, 10000}]; a[[2025]]

(2), 5202是第几个数？——56。——没有规律只能靠电脑。
FirstPosition[Union@IntegerReverse[Range@100^2], 5202]

5楼这串数没问题——3, 6, 19, 62, 195, 615, 1946, 6154, 19460, 61540, 194605, 615395, 1946050, 6153950, 19460499, 61539501, 194604990, 615395010, 1946049894, 6153950106, 19460498941, ——这是一串OEIS没有的数。

1位数有3个{1, 2, 3},

1,2位数有9个{1, 2, 3, 4, 5, 6, 7, 8, 9},
  
1,2,3位数有31个{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31},

1,2,3,4位数有99个{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, ......, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99},  

1,2,3,4,5位数有316个{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, ......, 308, 309, 310, 311,312, 313, 314, 315, 316}

最后一个数是这样一串数。{3, 9, 31, 99, 316, 999, 3162, 9999, 31622, 99999, 316227, 999999, 3162277, 9999999, 31622776, 99999999, 316227766, 999999999, 3162277660, 9999999999, 31622776601, 99999999999, 316227766016}
Table[Floor[10^(k/2) - Cos[k \[Pi]/2]^2], {k, 25}]

去掉个位数=0的{3, 9, 28, 90, 285, 900, 2846, 9000, 28460, 90000, 284605, 900000, 2846050, 9000000, 28460499, 90000000, 284604990, 900000000, 2846049894, 9000000000, 28460498941, 90000000000, 284604989415, 900000000000}
Table[Floor[10^(k/2) - Cos[k \[Pi]/2]^2] - Floor[(10^((k - 2)/2) - Cos[(k - 2) \[Pi]/2]^2)], {k, 25}]

后项-前项=3, 6, 19, 62, 195, 615, 1946, 6154, 19460, 61540, 194605, 615395, 1946050, 6153950, 19460499, 61539501, 194604990, 615395010, 1946049894, 6153950106, 19460498941, 61539501059, 194604989415, 615395010585,
Table[Floor[10^(k/2) - Cos[k \[Pi]/2]^2] - Floor[10^((k - 1)/2) - Cos[(k - 1) \[Pi]/2]^2] - Floor[10^((k - 2)/2) - Cos[(k - 2) \[Pi]/2]^2] + Floor[10^((k - 3)/2) - Cos[(k - 3) \[Pi]/2]^2], {k, 2, 48}]

化简可得Table[Floor[10^(k/2)] - Floor[10^((k - 1)/2)] - Floor[10^((k - 2)/2)] + Floor[10^((k - 3)/2)], {k, 2, 48}]

王守恩

5楼这串数没问题——3, 6, 19, 62, 195, 615, 1946, 6154, 19460, 61540, 194605, 615395, 1946050, 6153950, 19460499, 61539501, 194604990, 615395010, 1946049894, 6153950106, 19460498941, ——这是一串OEIS没有的数。

1位数+2位数=3+6=9, 19+62=81, 195+615=810, 1946+6154=8100, 19460+61540=81000, 194605+ 615395=810000, ......

{19, 195, 1946, 19460, 194605, 1946050, 19460499, 194604990, 1946049894, 19460498941, 194604989415, 1946049894152, 19460498941515, 194604989415154, 1946049894151542, 19460498941515414, 194604989415154140,

Table[Floor[Sqrt[10]^(2 k + 1)] - 9*10^(k - 1) - Floor[Sqrt[10]^(2 k - 1)], {k, 23}]

应该有更好的通项公式。

又。这些数没有规律——指的是个位数没有规律。首位数还是有规律的, 只能=1, 4, 5, 6, 9 五种可能。
{1, 4, 9,
18, 46, 52, 61, 63, 94,
121, 144, 148, 163, 169, 423, 441, 484, 487, 522, 526, 652, 675, 676, 691, 925, 927, 961, 982,
1042, 1062, 1089, 1251, 1273, 1297, 1405, 1426, 1656, 1674, 1828, 1843, 1861, 4032, 4069, 4072, 4201, 4264, 4276, 4441, 4477, 4483, 4633, 4648, 4671, 4806, 4815, 5202}

1位数=1(首位数1)+1(首位数4)+0(首位数5)+0(首位数6)+1(首位数9)=3个。
2位数=1(首位数1)+1(首位数4)+1(首位数5)+2(首位数6)+1(首位数9)=6个。
  合计=2(首位数1)+2(首位数4)+1(首位数5)+2(首位数6)+2(首位数9)=9个。
3位数=05(首位数1)+04(首位数4)+2(首位数5)+04(首位数6)+04(首位数9)=19个。
4位数=13(首位数1)+14(首位数4)+7(首位数5)+14(首位数6)+14(首位数9)=62个。
  合计=18(首位数1)+18(首位数4)+9(首位数5)+18(首位数6)+18(首位数9)=81个。
5位数=043(首位数1)+043(首位数4)+22(首位数5)+044(首位数6)+043(首位数9)=195个。
6位数=137(首位数1)+137(首位数4)+68(首位数5)+136(首位数6)+137(首位数9)=615个。
  合计=180(首位数1)+180(首位数4)+90(首位数5)+180(首位数6)+180(首位数9)=810个。
7位数=0432(首位数1)+0433(首位数4)+217(首位数5)+0432(首位数6)+0432(首位数9)=1946个。
8位数=1368(首位数1)+1367(首位数4)+683(首位数5)+1368(首位数6)+1368(首位数9)=6154个。
  合计=1800(首位数1)+1800(首位数4)+900(首位数5)+1800(首位数6)+1800(首位数9)=8100个。

northwolves

$a_k=\lfloor 10^k \left(\sqrt{10}-1\right)\rfloor -\lfloor 10^{k-1} \left(\sqrt{10}-1\right)\rfloor$