数字串的通项公式

王守恩

northwolves 发表于 2024-12-20 15:54
这两个是等价的
$\sum _{i=1}^{k+1} \lfloor \frac{n-i}{k-1}\rfloor -\sum _{i=1}^{n-k+1} \lfloor \frac{ ...

这些按钮我还是用不了。OEIS没有这串数。

周长相同的梯形。梯形={上底, 左腰, 右腰, 下底}=4个不同的整数=左腰<右腰。
a(1)=0
a(2)=0
a(3)=0
a(4)=0
a(5)=0
a(6)=0
a(7)=0
a(8)=0
a(9)=0,
a(10)=1=1,2,3,4,
a(11)=3=1,2,3,5=2,1,3,5=3,1,2,5,
a(12)=1=1,2,4,5,
a(13)=6=1,2,4,6=1,3,4,5=1,3,5,4=1,4,5,3=2,1,4,6=4,1,2,6,
a(14)=3=1,2,5,6=1,3,4,6=2,3,4,5,
a(15)=12=1,2,5,7=1,3,4,7=1,3,5,6=1,3,6,5=1,5,6,3=2,1,5,7=2,3,4,6=3,1,47=3,2,4,6=4,1,3,7=4,2,3,6=5,1,2,7,
a(16)=9=1,2,6,7=1,3,5,7=1,4,5,6=1,4,6,5=1,5,6,4=2,3,4,7=2,3,5,6=3,2,4,7=4,2,3,7,
a(17)=21=1268=1358=1367=1376=1457=1475=1574=1673=2168=2348=2357=2456=2465=2564=3158=3248=3257=4238=5138=5237=6128,
a(18)=12=1278=1368=1458=1467=1476=1674=2358=2367=2457=3258=3456=5238,
a(19)=33,
a(20)=24,
a(21)=48,
a(22)=34,
a(23)=69,
a(24)=52,
a(25)=
a(26)=
a(27)=
a(28)=

王守恩

已知梯形周长,  用4条整数边={上底,左腰,右腰,下底}={a,b,c,d},  来表示梯形最大面积。可以有公式(A)。

1. 公式(A)。Table[Maximize[{((d + a)Sqrt[(b + c - (d - a)) ((d - a) + b - c) (c + (d - a) - b) (b + c + (d - a))])/(4 (d - a)), a+b+c+d==n, 0<a<d, 0<b≤c, c-b<d-a<b+c},{a,b,c,d},Integers],{n,7,35}]

复制代码

a(07)={1,2,2,2}
a(08)={1,2,2,3}
a(09)={2,2,2,3}
a(10)={1,3,3,3}
a(11)={2,3,3,3}
a(12)={2,3,3,4}
a(13)={3,3,3,4}
a(14)={2,4,4,4}
a(15)={3,4,4,4}
a(16)={3,4,4,5}
a(17)={4,4,4,5}
a(18)={3,5,5,5}
a(19)={4,5,5,5}
a(20)={4,5,5,6}
a(21)={3,5,5,6}
a(22)={4,6,6,6}
a(23)={5,6,6,6}
a(24)={5,6,6,7}
a(25)={4,6,6,7}
a(26)={5,7,7,7}
a(27)={6,7,7,7}
a(28)={6,7,7,7}
a(29)={7,7,7,8}
公式(A)速度慢了。换成公式(B),答案没有变。

1. 公式(B)。Table[((n - 2 Floor[(n + 2)/4]) Sqrt[(2 Floor[(n + 4)/4] - n) (n - 4 Floor[(n + 2)/4] - 2 Floor[(n + 4)/4])])/4, {n, 7, 350}]

复制代码

对公式(B)取整数。得到公式(C)。

1. 公式(C)。Table[Round[((n - 2 Floor[(n + 2)/4]) Sqrt[(2 Floor[(n + 4)/4] - n) (n - 4 Floor[(n + 2)/4] - 2 Floor[(n + 4)/4])])/4], {n, 7, 350}]

复制代码

3, 3, 5, 6, 7, 8, 10, 12, 14, 15, 18, 20, 22, 24, 27, 30, 33, 35, 39, 42, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 85, 90, 95, 99, 105, 110, 115, 120, 126, 132, 138, 143, 150, 156, 162, 168, 175, 182, 189, 195,
203, 210, 217, 224, 232, 240, 248, 255, 264, 272, 280, 288, 297, 306, 315, 323, 333, 342, 351, 360, 370, 380, 390, 399, 410, 420, 430, 440, 451, 462, 473, 483, 495, 506, 517, 528, 540, 552, 564, 575, 588,
600, 612, 624, 637, 650, 663, 675, 689, 702, 715, 728, 742, 756, 770, 783, 798, 812, 826, 840, 855, 870, 885, 899, 915, 930, 945, 960, 976, 992, 1008, 1023, 1040, 1056, 1072, 1088, 1105, 1122, 1139, 1155,
1173, 1190, 1207, 1224, 1242, 1260, 1278, 1295, 1314, 1332, 1350, 1368, 1387, 1406, 1425, 1443, 1463, 1482, 1501, 1520, 1540, 1560, 1580, 1599, 1620, 1640, 1660, 1680, 1701, 1722, 1743, 1763, 1785, 1806}
单独把第1, 3, 5, 7, 9, ...项取出来。
3, 5, 7, 10, 14, 18, 22, 27, 33, 39, 45, 52, 60, 68, 76, 85, 95, 105, 115, 126, 138, 150, 162, 175, 189, 203, 217, 232, 248, 264, 280, 297, 315, 333, 351, 370, 390, 410, 430, 451, 473, 495, 517, 540, 564, 588,
612, 637, 663, 689, 715, 742, 770, 798, 826, 855, 885, 915, 945, 976, 1008, 1040, 1072, 1105, 1139, 1173, 1207, 1242, 1278, 1314, 1350, 1387, 1425, 1463, 1501, 1540, 1580, 1620, 1660, 1701, 1743, 1785,
可以有公式(D)。

1. 公式(D)。Table[Floor[(n/(1 + Power[E, (n)^-1]))^2], {n, 4, 100}]

复制代码

注意：公式(D)可是出现了大名鼎鼎的常数" e " !