1个编号是1的球,1个球的编号之和=1的整数倍的有1个。
2个编号是1的球,2个编号是2的球,从中取2个球,2个球的编号之和=2的整数倍的有2个。
3个编号是1的球,3个编号是2的球,3个编号是3的球,从中取3个球,3个球的编号之和=3的整数倍的有4个。
4个编号是1的球,4个编号是2的球,4个编号是3的球,4个编号是4的球,从中取4个球,4个球的编号之和=4的整数倍的有10个。
5个编号是1的球,5个编号是2的球,5个编号是3的球,5个编号是4的球,5个编号是5的球,从中取5个球,5个球的编号之和=5的整数倍的有26个。
6个编号是1的球,6个编号是2的球,6个编号是3的球,6个编号是4的球,6个编号是5的球,,6个编号是6的球,从中取6个球,6个球的编号之和=6的整数倍的有多少个?
7个编号是1的球,7个编号是2的球,7个编号是3的球,7个编号是4的球,7个编号是5的球,,7个编号是6的球,7个编号是7的球,从中取7个球,7个球的编号之和=7的整数倍的有多少个?
8个编号是1的球,8个编号是2的球,8个编号是3的球,8个编号是4的球,8个编号是5的球,,8个编号是6的球,8个编号是7的球,8个编号是8的球,从中取8个球,8个球的编号之和=8的整数倍的有多少个?
9个编号是1的球,9个编号是2的球,9个编号是3的球,9个编号是4的球,9个编号是5的球,,9个编号是6的球,9个编号是7的球,9个编号是8的球,9个编号是9的球,从中取9个球,9个球的编号之和=9的整数倍的有多少个?
王守恩 发表于 2025-5-3 07:39
1个编号是1的球,1个球的编号之和=1的整数倍的有1个。
2个编号是1的球,2个编号是2的球,从中取2个球,2个球的 ...
f=80
{80,{{1,1,1,1,1,1},{1,1,1,1,2,6},{1,1,1,1,3,5},{1,1,1,1,4,4},{1,1,1,2,2,5},{1,1,1,2,3,4},{1,1,1,3,3,3},{1,1,1,3,6,6},{1,1,1,4,5,6},{1,1,1,5,5,5},{1,1,2,2,2,4},{1,1,2,2,3,3},{1,1,2,2,6,6},{1,1,2,3,5,6},{1,1,2,4,4,6},{1,1,2,4,5,5},{1,1,3,3,4,6},{1,1,3,3,5,5},{1,1,3,4,4,5},{1,1,4,4,4,4},{1,1,4,6,6,6},{1,1,5,5,6,6},{1,2,2,2,2,3},{1,2,2,2,5,6},{1,2,2,3,4,6},{1,2,2,3,5,5},{1,2,2,4,4,5},{1,2,3,3,3,6},{1,2,3,3,4,5},{1,2,3,4,4,4},{1,2,3,6,6,6},{1,2,4,5,6,6},{1,2,5,5,5,6},{1,3,3,3,3,5},{1,3,3,3,4,4},{1,3,3,5,6,6},{1,3,4,4,6,6},{1,3,4,5,5,6},{1,3,5,5,5,5},{1,4,4,4,5,6},{1,4,4,5,5,5},{1,5,6,6,6,6},{2,2,2,2,2,2},{2,2,2,2,4,6},{2,2,2,2,5,5},{2,2,2,3,3,6},{2,2,2,3,4,5},{2,2,2,4,4,4},{2,2,2,6,6,6},{2,2,3,3,3,5},{2,2,3,3,4,4},{2,2,3,5,6,6},{2,2,4,4,6,6},{2,2,4,5,5,6},{2,2,5,5,5,5},{2,3,3,3,3,4},{2,3,3,4,6,6},{2,3,3,5,5,6},{2,3,4,4,5,6},{2,3,4,5,5,5},{2,4,4,4,4,6},{2,4,4,4,5,5},{2,4,6,6,6,6},{2,5,5,6,6,6},{3,3,3,3,3,3},{3,3,3,3,6,6},{3,3,3,4,5,6},{3,3,3,5,5,5},{3,3,4,4,4,6},{3,3,4,4,5,5},{3,3,6,6,6,6},{3,4,4,4,4,5},{3,4,5,6,6,6},{3,5,5,5,6,6},{4,4,4,4,4,4},{4,4,4,6,6,6},{4,4,5,5,6,6},{4,5,5,5,5,6},{5,5,5,5,5,5},{6,6,6,6,6,6}}}
f := Sum*Binomial/(2 n), {k, Divisors@n}]; Table, {n, 50}]
{1,2,4,10,26,80,246,810,2704,9252,32066,112720,400024,1432860,5170604,18784170,68635478,252088496,930138522,3446167860,12815663844,47820447028,178987624514,671825133648,2528212128776,9536895064400,36054433810102,136583761444364,518401146543812,1971076362005880,7506908923471954,28634752211620266,109385279303298134,418427080552561516,1602661111666612296,6146007503971075312,23596358977508462296,90692376956955350244,348936088066654523296,1343840109168425292660,5180299766448679532060,19987029597767017004376,77180849825857621779894,298278470246442673742468,1153638014306149018667804,4465167928718123725683748,17294692982395428197325698,67031948061107616221175888,259975635295723029619698462,1008913445455643197454196752}
A003239
Number of rooted planar trees with n non-root nodes: circularly cycling the subtrees at the root gives equivalent trees.
不错的一串数——A005920——有这串数——可惜没有这样的条文。
$(\frac{a_{1}+n}{a_{1}})(\frac{a_{2}+n}{a_{2}})(\frac{a_{3}+n}{a_{3}})(\frac{a_{4}+n}{a_{4}})\cdots(\frac{a_{n}+n}{a_{n}})=\frac{2n-1}{n}$
$a_{1},a_{2},a_{3},a_{4},\cdots,a_{n}=正整数.求:a_{1}+a_{2}+a_{3}+a_{4}+\cdots+a_{n}最小值$
$a_{1}+a_{2}+a_{3}+a_{4}+\cdots\cdots+a_{n}最小值=\frac{3 n^3 - 2 n^2 + n}{2}$
譬如。
a(1)=1,\(1=1,(\frac{1+0}{1})=\frac{1+0}{1}\)
a(2)=9,\(9=4+5,(\frac{4+1}{4})(\frac{5+1}{5})=\frac{2+1}{2}\)
a(3)=33,\(33=9+11+13,(\frac{9+2}{9})(\frac{11+2}{11})(\frac{13+2}{13})=\frac{3+2}{3}\)
a(4)=82,\(82=16+19+22+25,(\frac{16+3}{16})(\frac{19+3}{19})(\frac{22+3}{22})(\frac{25+3}{25})=\frac{4+3}{4}\)
a(5)=165,\(165=25+29+33+37+41,(\frac{25+4}{25})(\frac{29+4}{29})(\frac{33+4}{33})(\frac{37+4}{37})(\frac{41+4}{41})=\frac{5+4}{5}\)
a(6)=291,\(291=36+41+46+51+56+61,(\frac{36+5}{36})(\frac{41+5}{41})(\frac{46+5}{46})(\frac{51+5}{51})(\frac{56+5}{56})(\frac{61+5}{61})=\frac{6+5}{6}\)
a(7)=469,\(469=49+55+61+67+73+79+85,(\frac{49+6}{49})(\frac{55+6}{55})(\frac{61+6}{61})(\frac{67+6}{67})(\frac{73+6}{73})(\frac{79+6}{79})(\frac{85+6}{85})=\frac{7+6}{7}\)
a(8)=708,\(708=64+71+78+85+92+99+106+113,(\frac{64+7}{64})(\frac{71+7}{71})(\frac{78+7}{78})(\frac{85+7}{85})(\frac{92+7}{92})(\frac{99+7}{99})(\frac{106+7}{106})(\frac{113+7}{113})=\frac{8+7}{8}\)
a(9)=1017,\(1017=81+89+97+105+113+121+129+137+145,(\frac{81+8}{81})(\frac{89+8}{89})(\frac{97+8}{97})(\frac{105+8}{105})(\frac{113+8}{113})(\frac{121+8}{121})(\frac{129+8}{129})(\frac{137+8}{137})(\frac{145+8}{145})=\frac{9+8}{9}\)
{1, 9, 33, 82, 165, 291, 469, 708, 1017, 1405, 1881, 2454, 3133, 3927, 4845, 5896, 7089, 8433, 9937, 11610, 13461, 15499, 17733, 20172, 22825, 25701, 28809, 32158, 35757}
Table[(3 n^3 - 2 n^2 + n)/2, {n, 29}]
不错的一串数——还可以有更小的吗？
northwolves 发表于 2025-5-5 10:30
{1,2,4,10,26,80,246,810,2704,9252,32066,112720,400024,1432860,5170604,18784170,68635478,252088496, ...
Table, {n, 99}]
n可以是这串数吗——A208770——12, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 44, 45, 48, 52, 54, 55, 56, 60, 63, 65, 66, 70, 72, 75, 77, 78, 80, 84, 85, 88, 90, 91, 95, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 115, 117, 119, 120,
n是这串数——A208770——可有通项公式？谢谢！
n不是这串数——我们可有自己的通项公式？谢谢！！
已知：(n + 1)*a = n*a + 2 n + 1,a = 1, 求：通项公式。
王守恩 发表于 2025-5-13 14:56
已知：(n + 1)*a = n*a + 2 n + 1,a = 1, 求：通项公式。
b(n)=na(n),b(1)=1
b(n+1)=b(n)+2n+1
b(n)=n²
a(n)=n
有这样一串数——每次取2个数(允许重复取)相加,和可以跑遍所有偶数(2, 4, 6, 8, 10, 12, ..........)——还可以“稀”吗？
1, 3, 7, 9, 19, 21, 25, 27, 55, 57, 61, 63, 73, 75, 79, 81, 163, 165, 169, 171, 181, 183, 187, 189, 217, 219, 223, 225, 235, 237, 241, 243, 487, 489, 493, 495, 505, 507,
511, 513, 541, 543, 547, 549, 559, 561, 565, 567, 649, 651, 655, 657, 667, 669, 673, 675, 703, 705, 709, 711, 721, 723, 727, 729, 1459, 1461, 1465, 1467, 1477, ..........
A191106—有这串数———通项公式没我们的好——Table, 3] + 1, {n, 0, 64}]
northwolves 发表于 2025-5-13 15:55
b(n)=na(n),b(1)=1
b(n+1)=b(n)+2n+1
用 2，3，+，×，( )。譬如：
a(1)=0,
a(2)=1,{2},
a(3)=1,{3},
a(4)=2,{2*2},
a(5)=2,{2+3},
a(6)=2,{2*3},
a(7)=3,{2*2+3},
a(8)=3,{2*2*2},
a(9)=2,{3*3},
a(10)=3,{2(2+3)},
a(11)=3,{3*3+2},
a(12)=3,{3*3+3},
a(13)=4,{3*3+2*2}
a(14)=4,{3*3+2+3},
a(15)=3,{3(2+3},
a(16)=4,{2*2*2*2},
a(17)=4,{3(2+3)+2},
a(18)=3,{2*3*3},
a(19)=5,{2*2*2*2+3},
a(20)=4,{2*3*3+2},
a(21)=4,{2*3*3+3},
a(22)=4,{2(3*3+2)},
a(23)=5,{2*3*3+2+3}
a(24)=4,{2*3(2+2)},
a(25)=4,{(2+3)(2+3)},
a(26)=5,{2*3(2+2)+2},
a(27)=3,{3*3*3},
a(28)=5,{2(3*3+2+3)},
a(29)=4,{3*3*3+2},
a(30)=4,{3*3*3+3)},
a(31)=5,{3*3*3+2*2},
a(32)=5,{3*3*3+2+3},
a(33)=4,{3(3*3+2)}
a(34)=5,{2((3(2+3)+2)},
a(35)=5,{3(3*3+2)+2},
a(36)=4,{3(3*3+3)},
a(37)=6,{3(3*3+2)+2*2},
a(38)=5,{3(3*3+3)+2},
得到一串数：0, 1, 1, 2, 2, 2, 3, 3, 2, 3, 3, 3, 4, 4, 3, 4, 4, 3, 5, 4, 4, 4, 5, 4, 4, 5, 3, 5, 4, 4, 5, 5, 4, 5, 5, 4, 6, 5, ——OEIS没有这串数？——可有通项公式？谢谢！
第1个问题。
2^1-1!=1,
1*2^2-2!=2,
2*2^3-3!=10,
10*2^4-4!=136,
136*2^5-5!=4232,
4232*2^6-6!=270128,
270128*2^7-7!=34571344,
34571344*2^8-8!=8850223744,
......
得到一串数——{1, 2, 10, 136, 4232, 270128, 34571344, 8850223744, 4531314194048, 4640065731076352, 9502854617204452096, 38923692512068956783616, 318862889058868887744361472,
5224249574340507856716440066048, 171188210051989761448883000409892864, 11218990533967201006313996293939948847104, 1470495527268148970299588122238941287859519488}——求通项公式——问题1。
第2个问题。
2^1-1=1,
1*2^2-3=1,
1*2^3-6=2,
2*2^4-10=22,
22*2^5-15=689,
689*2^6-21=44075,
44075*2^7-28=5641572,
5641572*2^8-36=1444242396,
1444242396*2^9-45=739452106707,
.......
得到一串数——{1, 1, 2, 22, 689, 44075, 5641572, 1444242396, 739452106707, 757198957267913, 1550743464484685758, 6351845230529272864690, 52034316128495803307540389,
852530235449275241390741733271, 27935710755201851109891825115824008, 1830794740052908514337870650790642188152, 239965928168214824791293381940431052885458791}——求通项公式——问题2。
第3个问题。
136/22=6.18182,
4232/689=6.14224,
270128/44075=6.12883,
34571344/5641572=6.12796,
......
5224249574340507856716440066048/852530235449275241390741733271=6.12793465511211873361212958209,
......
262345168553487441312403456944635391129575147770993115164671264739968851694392029493592064/
42811352163266089318903999189948042642370639369565874377218194446248920891354503995981524=6.12793465511211873361208276053,
......
这个6.12793465511211873361208276053, 最后=？？？——问题3。