数字串的通项公式

王守恩

northwolves 发表于 2025-5-29 23:46

正三角形每边 1 等分可以有 0 个平行四边形。

正三角形每边 2 等分可以有 3 个平行四边形。

正三角形每边 3 等分可以有 15 个平行四边形。

正三角形每边 4 等分可以有  45  个平行四边形。

正三角形每边 5 等分可以有  105  个平行四边形。

正三角形每边 6 等分可以有  210  个平行四边形。

得到一串数——0, 3, 15, 45, 105, 210, 378, 630, 990, 1485, 2145, 3003, 4095, 5460, 7140, 9180, 11628, 14535, 17955, 21945, ......, n (n + 2) (n^2 - 1)/8,

正三角形每边 n 等分可以有  a(n)  个平行四边形——太可惜——A050534——就是没有这样的条文——你应该去补充一下——为广大数学爱好者。

王守恩

northwolves 发表于 2025-5-29 23:46

正六边形每边 1 等分可以有 6 个平行四边形。

正六边形每边 2 等分可以有 87 个平行四边形。

正六边形每边 3 等分可以有 417 个平行四边形。

正六边形每边 4 等分可以有  1278  个平行四边形。

正六边形每边 5 等分可以有  3060  个平行四边形。

正六边形每边 6 等分可以有  6261  个平行四边形。

得到一串数——6, 87, 417, 1278, 3060, 6261, 11487, 19452, 30978, 46995, 68541, 96762, 132912, 178353, 234555, 303096, 385662, ......, (9 n^4 + 4 n^3 - n)/2,

正六边形每边 n 等分可以有  a(n)  个平行四边形——太难了——A047786——没有——情有可原。

王守恩

northwolves 发表于 2025-5-29 23:46

由 2 个相同正三角形拼成平行四边形可以有 1 个平行四边形。

由 8 个相同正三角形拼成平行四边形可以有 13 个平行四边形。

由 18 个相同正三角形拼成平行四边形可以有 58 个平行四边形。

由 32 个相同正三角形拼成平行四边形可以有 170 个平行四边形。

由 50 个相同正三角形拼成平行四边形可以有 395 个平行四边形。

由 72 个相同正三角形拼成平行四边形可以有 791 个平行四边形。

得到一串数——1, 13, 58, 170, 395, 791, 1428, 2388, 3765, 5665, 8206, 11518, 15743, 21035, 27560, 35496, 45033, 56373, 69730, 85330, ......, n (n + 1) (3 n^2 + n - 1)/6,

由2*n^2个相同正三角形组成平行四边形可以有  a(n)  个平行四边形——简单的——A103220——就是没有这样的条文——有点说不过去了。

王守恩

a(1)=1,
a(2)=3=1+2,
a(3)=8=2+6=1+3+4,
a(4)=15=6+9=2+5+8=1+3+4+7,
a(5)=27=11+16=8+9+10=2+5+6+14=1+3+4+7+12,
a(6)=43=21+22=10+14+19=6+9+11+17=2+5+8+13+15=1+3+4+7+12+16,
a(7)=65=32+33=18+23+24=6+15+25+19=9+10+11+14+21=2+5+8+13+17+20=1+3+4+7+12+16+22,
a(8)=94=46+48=30+31+33=22+23+24+25=8+15+16+17+38=10+11+12+13+14+34=7+8+9+10+11+12+37=1+2+3+4+5+6+7+66,
a(9)=130=64+66=42+43+45=31+32+33+34=24+25+26+27+28=17+18+19+20+21+35=11+12+13+14+15+16+49=5+6+7+8+9+10+36+50=1+2+3+4+30+37+38+39+40,
a(10)=175=1+174=57+58+60=42+43+44+46=33+34+35+36+37=25+26+27+28+29+40=18+19+20+21+22+23+52=10+11+12+13+14+15+16+84=6+7+8+9+30+31+32+38+44=2+3+4+5+47+48+49+50+51+ 55,
a(11)=229=114+115=74+77+78=55+56+57+61=41+42+43+44+59=30+31+32+33+34+69=23+24+25+26+27+28+76=9+10+11+13+14+15+46+111
=2+5+6+8+17+18+19+20+134=21+47+48+49+50+51+52+53+54=1+3+4+7+12+16+22+29+36+45+54,
a(12)=294=146+148=97+98+99=72+73+74+75=56+58+59+60+61=40+41+42+43+44+84=33+34+35+37+38+39+78=24+25+26+27+28+30+31+103
=2+5+6+8+9+10+11+13+230=14+15+17+18+19+20+21+23+47+100==2+10+11+13+17+23+30+37+46+55+66=1+3+4+7+12+16+22+29+36+45+54+65,

{1, 3, 8, 15, 27, 43, 65, 94, 130, 175, 229, 294, 369, 456, 557, 671, 800, 944, 1105, 1283, 1479, 1695, 1930, 2187, 2465, 2765, 3090, 3439, 3813, 4213, 4641, 5096, 5580, 6095, 6639, 7216, 7825, 8466, 9143, 9855, 10602,

Table[SeriesCoefficient[(x + 3 x^3 - 2 x^4 + 6 x^5 - 5 x^6 + 9 x^7 - 7 x^8 + 10 x^9 - 7 x^10 + 9 x^11 - 5 x^12 + 5 x^13 - 3 x^14 + 3 x^15 - x^16)/((1 - x)^2 (1 - x^3)^2 (1 + x^6) (1 - x + x^2)), {x, 0, n}], {n, 100}]

a(7)=65=32+33=20+21+24=14+15+17+19=9+10+11+12+23=5+6+7+13+16+18=1+2+3+4+8+22+25,
a(7)=65=32+33=18+23+24=6+15+25+19=9+10+11+14+21=2+5+8+13+17+20=1+3+4+7+12+16+22,

补充内容 (2025-6-7 06:51):
没用删了，删不了。

王守恩

a(1)=1,
a(2)=3=1+2,
a(3)=8=2+6=1+3+4,
a(4)=15=6+9=2+5+8=1+3+4+7,
a(5)=27=11+16=8+9+10=2+5+6+14=1+3+4+7+12,
a(6)=43=21+22=10+14+19=6+9+11+17=2+5+8+13+15=1+3+4+7+12+16,
a(7)=65=32+33=18+23+24=6+15+25+19=9+10+11+14+21=2+5+8+13+17+20=1+3+4+7+12+16+22,
a(8)=94=46+48=30+31+33=22+23+24+25=8+15+16+17+38=10+11+12+13+14+34=7+8+9+10+11+12+37=1+2+3+4+5+6+7+66,
a(9)=130=64+66=42+43+45=31+32+33+34=24+25+26+27+28=17+18+19+20+21+35=11+12+13+14+15+16+49=5+6+7+8+9+10+36+50=1+2+3+4+30+37+38+39+40,
a(10)=175=1+174=57+58+60=42+43+44+46=33+34+35+36+37=25+26+27+28+29+40=18+19+20+21+22+23+52=10+11+12+13+14+15+16+84=6+7+8+9+30+31+32+38+44=2+3+4+5+47+48+49+50+51+ 55,
a(11)=229=114+115=74+77+78=55+56+57+61=41+42+43+44+59=30+31+32+33+34+69=23+24+25+26+27+28+76=9+10+11+13+14+15+46+111
=2+5+6+8+17+18+19+20+134=21+47+48+49+50+51+52+53+54=1+3+4+7+12+16+22+29+36+45+54,
a(12)=294=146+148=97+98+99=72+73+74+75=56+58+59+60+61=40+41+42+43+44+84=33+34+35+37+38+39+78=24+25+26+27+28+30+31+103
=2+5+6+8+9+10+11+13+230=14+15+17+18+19+20+21+23+47+100==2+10+11+13+17+23+30+37+46+55+66=1+3+4+7+12+16+22+29+36+45+54+65,

{1, 3, 8, 15, 27, 43, 65, 94, 130, 175, 229, 294, 369, 456, 557, 671, 800, 944, 1105, 1283, 1479, 1695, 1930, 2187, 2465, 2765, 3090, 3439, 3813, 4213, 4641, 5096, 5580, 6095, 6639, 7216, 7825, 8466, 9143, 9855, 10602,

Table[SeriesCoefficient[(x + 3 x^3 - 2 x^4 + 6 x^5 - 5 x^6 + 9 x^7 - 7 x^8 + 10 x^9 - 7 x^10 + 9 x^11 - 5 x^12 + 5 x^13 - 3 x^14 + 3 x^15 - x^16)/((1 - x)^2 (1 - x^3)^2 (1 + x^6) (1 - x + x^2)), {x, 0, n}], {n, 100}]

王守恩

northwolves 发表于 2025-5-29 23:46

再找一串不就得了——用 1, 2, 3, +, ×, ( )。
a(1)=1,{1},
a(2)=1,{2},
a(3)=1,{3},
a(4)=2,{1+3},
a(5)=2,{2+3},
a(6)=2,{3+3},
a(7)=3,{1+6},
a(8)=3,{2*4},
a(9)=2,{3*3},
a(10)=3,{1+9},
a(11)=3,{2+9},
a(12)=3,{3+9},
a(13)=4,{1+12},
a(14)=4,{2+12},
a(15)=3,{3*5},
a(16)=4,{2*8},
a(17)=4,{1+16},
a(18)=3,{2*9},
a(19)=4,{1+18},
a(20)=4,{2+18},
a(21)=4,{3+18},
a(22)=4,{2*11},
a(23)=5,{1+22},
a(24)=4,{3*8},
a(25)=4,{5*5},
a(26)=5,{2*13},
a(27)=3,{3*9},
a(28)=4,{1+27},
a(29)=4,{2+27},
a(30)=4,{3*10},

第1个数 = 1, 4, 7, 13, 23, 41, 71, 131, 239, 593, 719, 1931, 3779, 9059, ——清一色的素数。

王守恩

编号为1,2,3,...,n的n个球,  重量(正整数)依次为f(1)<f(2)<f(3)<...<f(n)。

选若干数目的球并将所选的球分成两堆,无论哪堆有哪些球,一定满足以下条件:

①若两堆球的数目不同,则球较多的一堆一定比另一堆重。

②若两堆球的数目相同,则两堆中编号最大球所在的一堆一定比另一堆重。

求a(n)=f(1)+f(2)+f(3)+...+f(n)的最小值。譬如:
a(1)=1=1,
a(2)=1+2=3,
a(3)=2+3+4=9,
a(4)=4+5+6+8=23,
a(5)=9+10+11+13+16=59,
a(6)=18+19+20+22+25+31=135,
a(7)=38+39+40+42+45+51+62=317,
a(8)=77+78+79+81+84+90+101+123=713,
a(9)=158+159+160+162+165+171+182+204+246=1607,

参考的是这串数——A062178——a(n+1) = 2a(n)-a([n/2]) starting with a(0)=0 and a(1)=1.

0, 1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639, 83189, 166289, 332405, 664637, 1328936, 2657534, 5314400, 10628132, 21254942, 42508562,

a(1)=1=1,{1}+0,
a(2)=1+2=3,{1,2}+0,
a(3)=2+3+4=9,{1,2,3}+1,
a(4)=4+5+6+8=23,{1,2,3,5}+1,
a(5)=9+10+11+13+16=59,{1,2,3,5,8}+8,
a(6)=18+19+20+22+25+31=135,{1,2,3,5,8,14}+17,
a(7)=38+39+40+42+45+51+62=317,{1,2,3,5,8,14,25}+37,
a(8)=77+78+79+81+84+90+101+123=713,{1,2,3,5,8,14,25,47}+76,
a(9)=158+159+160+162+165+171+182+204+246=1607,{1,2,3,5,8,14,25,47,89}+157,

a (1) = {1} + 0,
a (2) = {1, 2} + 0,
a (3) = {1, 2, 3} + 1,
a (4) = {1, 2, 3, 5} + 3,
a (5) = {1, 2, 3, 5, 8} + 8,
a (6) = {1, 2, 3, 5, 8, 14} + 17,
a (7) = {1, 2, 3, 5, 8, 14, 25} + 37,
a (8) = {1, 2, 3, 5, 8, 14, 25, 47} + 76,
a (9) = {1, 2, 3, 5, 8, 14, 25, 47, 89} + 157,
a (10) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173} + 316,
a (11) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338} + 640,
a (12) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668} + 1283,
a (13) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322} + 2580,
a (14) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630} + 5163,
a (15) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235} + 10351,
a (16) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445} + 20707,
a (17) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843} + 41461,
a (18) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639} + 82927,
a (19) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639, 83189} + 165943,
a (20) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639, 83189, 166289} + 331894,
a (21) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639, 83189, 166289, 332405} + 663961,
a (22) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639, 83189, 166289, 332405, 664637} + 1327930,
a (23) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639, 83189, 166289, 332405, 664637, 1328936} + 2656198,
a (24) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639, 83189, 166289, 332405, 664637, 1328936, 2657534} + 5312410,

求助！最后一个数什么规律？谢谢！！！

王守恩

接楼上。

a (1) = 1 = {1} + 1 (2 - 1 - 1) = 1,
a (2) = 1 + 2 = {1 + 2} + 2 (3 - 1 - 2) = 3,
a (3) = 2 + 3 + 4 = {1 + 2 + 3} + 3 (5 - 2 - 2) = 9,
a (4) = 4 + 5 + 6 + 8 = {1 + 2 + 3 + 5} + 4 (8 - 2 - 3) = 23,
a (5) = 9 + 10 + 11 + 13 + 16 = {1 + 2 + 3 + 5 + 8} + 5 (14 - 3 - 3) = 59,
a (6) = 18 + 19 + 20 + 22 + 25 + 31 = {1 + 2 + 3 + 5 + 8 + 14} + 6 (25 - 3 - 5) = 135,
a (7) = 38 + 39 + 40 + 42 + 45 + 51 + 62 = {1 + 2 + 3 + 5 + 8 + 14 + 25} + 7 (47 - 5 - 5) = 317,
a (8) = 77 + 78 + 79 + 81 + 84 + 90 + 101 + 123 = {1, 2, 3, 5, 8, 14, 25, 47} + 8 (89 - 5 - 8) = 713,
a (9) = 158 + 159 + 160 + 162 + 165 + 171 + 182 + 204 + 246 = {1, 2, 3, 5, 8, 14, 25, 47, 89} + 9 (173 - 8 - 8) = 1607,
a (10) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173} + 10 (338 - 8 - 14) = 3527,
a (11) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338} + 11 (668 - 14 - 14) = 7745,
a (12) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668} + 12 (1322 - 14 - 25) = 16769,
a (13) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322} + 13 (2630 - 25 - 25) = 36235,
a (14) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630} + 14 (5235 - 25 - 47) = 77607,
a (15) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235} + 15 (10445 - 47 - 47) = 165825,
a (16) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445} + 16 (20843 - 47 - 89) = 352317,
a (17) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843} + 17 (41639 - 89 - 89) = 746685,
a (18) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639} + 18 (83189 - 89 - 173) = 1576173,
a (19) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639, 83189} + 19 (166289 - 173 - 173) = 3319593,
a (20) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639, 83189, 166289} + 20 (332405 - 173 - 338) = 6970845,
a (21) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639, 83189, 166289, 332405} + 21 (664637 - 338 - 338) = 14608551,
a (22) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639, 83189, 166289, 332405, 664637} + 22 (1328936 - 338 - 668) = 30544467,
a (23) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639, 83189, 166289, 332405, 664637, 1328936} + 23 (2657534 - 668 - 668) = 63751497,
a (24) = {1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639, 83189, 166289, 332405, 664637, 1328936, 2657534} + 24 (5314400 - 668 - 1322) = 132814317,
                    
求助！最后一个数什么规律？谢谢！！！

王守恩

A206226——1, 3, 12, 64, 377, 2432, 16475, 116263, 845105, 6292069, 47759392, 368379006, 2879998966, 22777018771, 181938716422, 1465972415692, 11902724768574,

Table[Length@IntegerPartitions[n^2, n], {n, 29}]——这个通项公式太慢了。

王守恩

接708#——编号为 1, 2, 3, ..., n 的 n 个球,  重量(正整数)依次为 f(1) < f(2) < f(3) < ... < f(n)。

选若干数目的球并将所选的球分成两堆,无论哪堆有哪些球,一定满足以下条件:

① 若两堆球的数目不同,则球较多的一堆一定比另一堆重。

② 若两堆球的数目相同,则两堆中编号最大球所在的一堆一定比另一堆重。

求 a(n) = f(1) + f(2) + f(3) + ... + f(n) 的最小值。

a (1) = 1 = {1} + 1 (2 - 1 - 1) = 1,
a (2) = 1 + 2 = {1 + 2} + 2 (3 - 1 - 2) = 3,
a (3) = 2 + 3 + 4 = {1 + 2 + 3} + 3 (5 - 2 - 2) = 9,
a (4) = 4 + 5 + 6 + 8 = {1 + 2 + 3 + 5} + 4 (8 - 2 - 3) = 23,
a (5) = 9 + 10 + 11 + 13 + 16 = {1 + 2 + 3 + 5 + 8} + 5 (14 - 3 - 3) = 59,
a (6) = 18 + 19 + 20 + 22 + 25 + 31 = {1 + 2 + 3 + 5 + 8 + 14} + 6 (25 - 3 - 5) = 135,
a (7) = 38 + 39 + 40 + 42 + 45 + 51 + 62 = {1 + 2 + 3 + 5 + 8 + 14 + 25} + 7 (47 - 5 - 5) = 317,
a (8) = 77 + 78 + 79 + 81 + 84 + 90 + 101 + 123 = {1 + 2 + 3 + 5 + 8 + 14 + 25 + 47} + 8 (89 - 5 - 8) = 713,
a (9) = 158 + 159 + 160 + 162 + 165 + 171 + 182 + 204 + 246 = {1 + 2 + 3 + 5 + 8 + 14 + 25 + 47 + 89} + 9 (173 - 8 - 8) = 1607,

得到一串数:  1, 3, 9, 23, 59, 135, 317, 713, 1607, 3527, 7745, 16769, 36235, 77607, 165825, 352317, 746685, 1576173, 3319593, 6970845, 14608551, 30544467, 63751497, 132814317,

\(\D a(n)=\sum_{k=1}^nb_{k}+n\cdot c(n)=\sum_{k=1}^nb_{k}+n\cdot\bigg(b_{n+1}-b_{\lfloor(n+1)/2\rfloor}-b_{\lfloor(n+2)/2\rfloor}\bigg)\)

$b_{k}$参考的是这串数——A062178——0, 1, 2, 3, 5, 8, 14, 25, 47, 89, 173, 338, 668, 1322, 2630, 5235, 10445, 20843, 41639, 83189, 166289, 332405, 664637, 1328936, 2657534, 5314400, 10628132, 21254942, 42508562,

\(\D b_{0}=0,b_{1}=1,b_{n+1}=2\cdot b_{n}-b_{\lfloor n/2\rfloor},n≥1\)

各位可有一次到位的通项公式？谢谢！！！