Pure problem

Ansette · 2006-03-05 03:40:44

We've been givin a problem involving boxes (parcels basically)..

The dimensions of each parcel are integers abc with 1≤a≤b≤c≤n

Basically such that for each n, there are many combinations as you can see, for instance with n=2, the options are:

1,1,1
1,1,2
1,2,2
2,2,2

to fit within those bounds.

The first thing we've noticed is that the number of parcels per each n is a tetrahedral number.. e.g. general formula with our n = n(n+1)(n+2)/6

e.g. 1,4,10,20,35.. etc

The next thing of note is that the total volume for each set of parcels fits the pattern of is 1,15,90,350,1050.. Stirling numbers of second order so i'm informed by the internet..

What I'm trying to ask is for an explanation as to why this is the case?
(And also a general formula for stirling numbers of second order if possible!)

krassi_holmz · 2006-03-05 04:19:12

First look at 2d.
It will help you with the 3d.

krassi_holmz · 2006-03-05 04:27:10

Let

is the number of all sets of k elements

for which

Last edited by krassi_holmz (2006-03-05 04:28:43)

krassi_holmz · 2006-03-05 04:31:05

How can we compute

?
I noticed this:

krassi_holmz · 2006-03-05 04:39:06

Cool!!!
Here's a proof:
for some x the number of all sets

--->(1)
for which

is exactly
If we summarize (1) for all x≤n then we'll get the number or all sets for which

which is exacly

Ansette · 2006-03-05 04:46:11

i have to admit i'm slightly confused..

krassi_holmz · 2006-03-05 04:59:09

It's not hard to consider that

.
So

so here's your explanation why the number is tetrahedral number.

Ansette · 2006-03-05 05:07:49

ah of course.. working like the binomial theorem..!!

krassi_holmz · 2006-03-05 05:13:43

It can be shown with simple induction that:

krassi_holmz · 2006-03-05 05:16:23

This was the complete examination of your first problem.
The second seems much interesting for me.
Now I'll search the web for these stirling numbers.

krassi_holmz · 2006-03-05 05:32:30

After some web search I can tell you that if we name your sequence with a:
a1=1
a2=15
a3=90
a4=350
a5=1050

It seems that:

krassi_holmz · 2006-03-05 05:39:26

With some simplification it seems that the formula is:

, where is the binomial coeficient.

Last edited by krassi_holmz (2006-03-05 05:42:09)

krassi_holmz · 2006-03-05 05:46:31

I tried and it seems that the above formula is correct.
I don't know why but it's true.
It will be hard task to simplify it more.
That was very good question!!!

Ansette · 2006-03-05 06:04:47

ok im very impressed lol.. i had some trouble making the formula work at first.. misread the last j as a 3 so was writing iC3 each time.. really math it up lol..

there is a third question really..

we have to try and place these individual boxes into a large parcel that must be rectangular with no free space, though the dimensions are irrelevant.. kind of like a three-dimensional tetris..

any thoughts?

(alternatively if one parcel is not possible, as few as possible..)

obviously for n=1, 1 package
for n=2 it requires 2, but then that's understandable with a volume of 15.
for n=3 i believe we've managed 1, but after that we're unsure

Last edited by Ansette (2006-03-05 06:07:16)

krassi_holmz · 2006-03-05 06:17:16

Good.
Can I ask where are this questions from?
There are very interesting but...
very hard too.
The third is the hardest.
My suggestion: try solving it in 2d first. I'll try too.
I'll have a closer look at a_i.

krassi_holmz · 2006-03-05 06:18:55

Here are the first 50 terms of a_i:
{1, 15, 90, 350, 1050, 2646, 5880, 11880, 22275, 39325, 66066, 106470,
165620, 249900, 367200, 527136, 741285, 1023435, 1389850, 1859550, 2454606,
3200450, 4126200, 5265000, 6654375, 8336601, 10359090, 12774790, 15642600,
19027800, 23002496, 27646080, 33045705, 39296775, 46503450, 54779166,
64247170, 75041070, 87305400, 101196200, 116881611, 134542485, 154373010,
176581350, 201390300, 229037956, 259778400, 293882400, 331638125, 373351875}

Ansette · 2006-03-05 06:24:36

we have a presentation on this, first year group work basically (designed no doubt in a way to build team skills..) we'd noticed a lot of the patterns between us however were unsure what to make of them or how to prove them :s

krassi_holmz · 2006-03-05 06:26:01

And here's a set of the divisors of each (you may need it ):
{{1}, {1, 3,
5, 15}, {1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90}, {1, 2, 5, 7, 10, 14,
25, 35, 50, 70, 175, 350}, {1, 2, 3, 5, 6,
7, 10, 14, 15, 21, 25, 30, 35, 42, 50, 70, 75, 105, 150, 175, 210, 350,
525, 1050}, {1, 2, 3, 6,
7, 9, 14, 18, 21, 27, 42, 49, 54, 63, 98, 126, 147, 189, 294, 378, 441, 882,
1323, 2646}, {1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28,
30, 35, 40, 42, 49, 56, 60, 70, 84, 98, 105, 120, 140, 147, 168, 196,
210, 245, 280, 294, 392, 420, 490, 588, 735, 840, 980, 1176, 1470,
1960, 2940, 5880}, {1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 18,
20, 22, 24, 27, 30, 33, 36, 40, 44, 45, 54, 55, 60, 66, 72, 88, 90, 99,
108, 110, 120, 132, 135, 165, 180, 198, 216, 220, 264, 270, 297, 330, 360,
396, 440, 495, 540, 594, 660, 792, 990, 1080, 1188, 1320, 1485, 1980,
2376, 2970, 3960, 5940, 11880}, {1, 3, 5, 9, 11, 15, 25, 27, 33, 45, 55,
75, 81, 99, 135, 165, 225, 275, 297, 405, 495, 675, 825, 891,
1485, 2025, 2475, 4455, 7425, 22275}, {1, 5, 11, 13, 25, 55, 65,
121, 143, 275, 325, 605, 715, 1573, 3025, 3575, 7865, 39325}, {1, 2, 3, 6,
7, 11, 13, 14,
21, 22, 26, 33, 39, 42, 66, 77, 78, 91, 121, 143, 154, 182, 231, 242,
273, 286, 363, 429, 462, 546, 726, 847, 858, 1001, 1573, 1694, 2002, 2541,
3003, 3146, 4719, 5082, 6006, 9438, 11011, 22022, 33033, 66066}, {1, 2,
3, 5, 6, 7, 9, 10, 13, 14, 15, 18, 21, 26, 30, 35, 39, 42, 45, 63, 65,
70, 78, 90, 91, 105, 117, 126, 130, 169, 182, 195, 210, 234, 273, 315,
338, 390, 455, 507, 546, 585, 630, 819, 845, 910, 1014, 1170, 1183, 1365,
1521, 1638,
1690, 2366, 2535, 2730, 3042, 3549, 4095, 5070, 5915, 7098, 7605, 8190,
10647, 11830, 15210, 17745, 21294, 35490, 53235, 106470}, {1, 2, 4, 5, 7,
10, 13, 14, 20, 26, 28, 35, 49, 52, 65, 70, 91, 98, 130, 140, 169, 182,
196, 245, 260, 338, 364, 455, 490, 637, 676, 845, 910, 980, 1183, 1274,
1690, 1820, 2366, 2548, 3185, 3380, 4732, 5915, 6370, 8281, 11830, 12740,
16562, 23660, 33124, 41405, 82810,
165620}, {1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 17, 20, 21, 25, 28, 30, 34,
35, 42, 49, 50, 51, 60, 68, 70, 75, 84, 85, 98, 100, 102, 105, 119,
140, 147, 150, 170, 175, 196, 204, 210, 238, 245, 255, 294, 300, 340,
350, 357, 420, 425, 476, 490, 510, 525, 588, 595, 700, 714, 735, 833,
850, 980, 1020, 1050, 1190, 1225, 1275, 1428, 1470, 1666, 1700, 1785,
2100, 2380, 2450, 2499, 2550, 2940, 2975, 3332, 3570, 3675,
4165, 4900, 4998, 5100, 5950, 7140, 7350, 8330, 8925, 9996, 11900, 12495,
14700, 16660, 17850, 20825, 24990, 35700, 41650, 49980, 62475, 83300,
124950, 249900}, {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 17, 18, 20, 24,
25, 27, 30, 32, 34, 36, 40, 45, 48, 50, 51, 54, 60, 68, 72, 75, 80, 85,
90, 96, 100, 102, 108, 120, 135, 136, 144, 150, 153, 160, 170, 180, 200,
204, 216, 225, 240, 255, 270, 272, 288, 300, 306, 340, 360, 400, 408, 425
, 432, 450, 459, 480, 510, 540, 544, 600, 612, 675, 680, 720, 765,
800, 816, 850, 864, 900, 918, 1020, 1080, 1200, 1224, 1275, 1350, 1360,
1440, 1530, 1632, 1700, 1800, 1836, 2040, 2160, 2295, 2400, 2448, 2550,
2700, 2720, 3060, 3400, 3600, 3672, 3825, 4080, 4320, 4590, 4896, 5100,
5400, 6120, 6800, 7200, 7344, 7650, 8160, 9180, 10200, 10800, 11475,
12240, 13600, 14688, 15300, 18360, 20400, 21600, 22950, 24480, 30600,
36720, 40800, 45900, 61200, 73440, 91800, 122400, 183600, 367200}, {
1, 2, 3, 4, 6, 8, 12, 16, 17, 19, 24, 32, 34, 38, 48, 51, 57, 68, 76,
96, 102, 114, 136, 152, 204, 228, 272, 289, 304, 323, 408, 456, 544,
578, 608, 646, 816, 867, 912, 969, 1156, 1292, 1632, 1734, 1824, 1938,
2312, 2584, 3468, 3876, 4624, 5168, 5491, 6936, 7752, 9248, 10336, 10982,
13872, 15504, 16473, 21964, 27744, 31008, 32946, 43928, 65892,
87856, 131784, 175712, 263568, 527136}, {1, 3, 5, 9, 15, 17, 19, 27, 45,
51, 57, 85, 95, 135, 153, 171, 255, 285, 289, 323, 459, 513, 765,
855, 867, 969, 1445, 1615, 2295, 2565, 2601, 2907, 4335, 4845, 5491, 7803,
8721, 13005, 14535, 16473, 27455, 39015, 43605, 49419, 82365, 148257,
247095, 741285}, {1, 3, 5, 7, 9, 15, 19, 21, 27, 35, 45, 57, 63, 81,
95, 105, 133, 135, 171, 189, 285, 315, 361, 399, 405, 513, 567, 665,
855, 945, 1083, 1197, 1539, 1805, 1995, 2527, 2565, 2835, 3249,
3591, 5415, 5985, 7581, 7695, 9747, 10773, 12635, 16245, 17955, 22743,
29241, 37905, 48735, 53865, 68229, 113715, 146205, 204687,
341145, 1023435}, {1, 2, 5, 7, 10, 11, 14, 19, 22, 25, 35, 38, 50,
55, 70, 77, 95, 110, 133, 154, 175, 190, 209, 266, 275, 350,
361, 385, 418, 475, 550, 665, 722, 770,
950, 1045, 1330, 1463, 1805, 1925, 2090, 2527, 2926, 3325, 3610, 3850,
3971, 5054, 5225, 6650, 7315, 7942, 9025, 10450, 12635,
14630, 18050, 19855, 25270, 27797, 36575, 39710, 55594, 63175, 73150,
99275, 126350, 138985, 198550, 277970, 694925, 1389850}, {1,
2, 3, 5, 6, 7, 10, 11, 14, 15, 21, 22, 23, 25, 30, 33, 35, 42,
46, 49, 50, 55, 66, 69, 70, 75, 77, 98, 105, 110, 115, 138, 147, 150,
154, 161, 165, 175, 210, 230, 231, 245, 253, 275, 294, 322, 330, 345, 350,
385, 462, 483, 490, 506, 525, 539, 550, 575, 690, 735, 759, 770, 805, 825,
966, 1050, 1078, 1127, 1150, 1155, 1225, 1265, 1470, 1518, 1610,
1617, 1650, 1725, 1771, 1925, 2254, 2310, 2415, 2450, 2530, 2695, 3234,
3381, 3450, 3542, 3675, 3795, 3850, 4025, 4830, 5313, 5390, 5635, 5775,
6325, 6762, 7350, 7590, 8050, 8085, 8855, 10626, 11270, 11550, 12075,
12397, 12650, 13475, 16170,
16905, 17710, 18975, 24150, 24794, 26565, 26950, 28175, 33810, 37191,
37950, 40425, 44275, 53130, 56350, 61985, 74382, 80850, 84525, 88550, 123970,
132825, 169050, 185955, 265650, 309925, 371910, 619850,
929775, 1859550}, {1, 2, 3, 6, 7, 9, 11, 14, 18, 21, 22, 23, 33, 42, 46,
49, 63, 66, 69, 77, 98, 99, 121, 126, 138, 147, 154, 161, 198,
207, 231, 242, 253, 294, 322, 363, 414, 441, 462, 483, 506, 539, 693,
726, 759, 847, 882, 966, 1078, 1089, 1127, 1386, 1449, 1518, 1617, 1694,
1771, 2178, 2254, 2277, 2541, 2783, 2898, 3234,
3381, 3542, 4554, 4851, 5082, 5313,
5566, 5929, 6762, 7623, 8349, 9702, 10143, 10626, 11858, 12397, 15246,
15939, 16698, 17787, 19481, 20286, 24794, 25047, 31878, 35574, 37191,
38962, 50094, 53361, 58443, 74382, 106722, 111573, 116886, 136367, 175329,
223146, 272734, 350658, 409101, 818202, 1227303, 2454606}, {1, 2, 5,
10, 11, 22, 23, 25, 46, 50, 55, 110, 115, 121, 230, 242, 253, 275, 506,
529, 550, 575, 605, 1058, 1150, 1210, 1265, 2530, 2645, 2783, 3025, 5290,
5566, 5819, 6050, 6325, 11638, 12650, 13225, 13915, 26450, 27830, 29095,
58190, 64009, 69575, 128018, 139150, 145475, 290950, 320045, 640090,
1600225, 3200450}, {1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 15, 20, 23, 24, 25,
26, 30, 39, 40, 46, 50, 52, 60, 65, 69, 75, 78, 92, 100, 104, 115, 120,
130, 138, 150, 156, 184, 195, 200, 230, 260, 276, 299, 300, 312, 325, 345,
390, 460, 520, 529, 552, 575, 598, 600, 650, 690, 780, 897, 920, 975,
1058, 1150, 1196, 1300, 1380, 1495, 1560, 1587, 1725, 1794, 1950, 2116,
2300, 2392, 2600, 2645, 2760, 2990, 3174, 3450, 3588, 3900, 4232, 4485,
4600, 5290, 5980, 6348, 6877, 6900, 7176, 7475, 7800, 7935, 8970, 10580,
11960, 12696, 13225, 13754, 13800, 14950, 15870, 17940, 20631, 21160,
22425, 26450, 27508, 29900, 31740, 34385, 35880, 39675, 41262, 44850,
52900, 55016, 59800, 63480, 68770, 79350, 82524, 89700, 103155, 105800,
137540, 158700, 165048, 171925, 179400, 206310,
275080, 317400, 343850, 412620, 515775, 687700, 825240, 1031550, 1375400,
2063100, 4126200}, {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 18, 20, 24,
25, 26, 27, 30, 36, 39, 40, 45, 50, 52, 54, 60, 65, 72, 75, 78, 81, 90,
100, 104, 108, 117, 120, 125, 130, 135, 150, 156, 162, 180, 195,
200, 216, 225, 234, 250, 260, 270, 300, 312, 324, 325, 351, 360, 375,
390, 405, 450, 468, 500, 520, 540, 585, 600, 625, 648, 650, 675, 702, 750,
780, 810, 900, 936, 975, 1000, 1053, 1080, 1125, 1170, 1250, 1300,
1350, 1404, 1500, 1560, 1620, 1625, 1755, 1800, 1875, 1950, 2025, 2106,
2250, 2340, 2500, 2600, 2700, 2808, 2925, 3000, 3240, 3250, 3375, 3510,
3750, 3900, 4050, 4212, 4500, 4680, 4875, 5000, 5265, 5400, 5625,
5850, 6500, 6750, 7020, 7500, 7800, 8100, 8125, 8424, 8775, 9000, 9750,
10125, 10530, 11250, 11700, 13000, 13500, 14040, 14625, 15000, 16200,
16250, 16875, 17550, 19500, 20250, 21060, 22500, 23400, 24375,
26325, 27000, 29250, 32500, 33750, 35100, 39000, 40500, 42120,
43875, 45000, 48750, 50625, 52650, 58500, 65000, 67500, 70200, 73125,
81000, 87750, 97500, 101250,
105300, 117000, 131625, 135000, 146250, 175500, 195000, 202500, 210600,
219375, 263250, 292500, 351000, 405000, 438750, 526500, 585000, 658125,
877500, 1053000, 1316250, 1755000, 2632500, 5265000}, {1, 3, 5, 7, 9, 13, 15,
21, 25, 35, 39, 45, 63, 65, 75, 91, 105, 117, 125, 169, 175,
195, 225, 273, 315, 325, 375, 455, 507, 525, 585, 625, 819, 845, 875,
975, 1125, 1183, 1365, 1521, 1575, 1625, 1875, 2275,
2535, 2625, 2925, 3549, 4095, 4225, 4375, 4875, 5625, 5915, 6825,
7605, 7875, 8125, 10647, 11375, 12675, 13125, 14625, 17745, 20475, 21125,
24375, 29575, 34125, 38025, 39375, 53235, 56875, 63375, 73125, 88725,
102375, 105625, 147875, 170625, 190125, 266175,
316875, 443625, 511875, 739375, 950625, 1330875, 2218125, 6654375}, {1,
3, 7, 9, 13, 21, 27, 29, 39, 63, 81, 87, 91, 117, 169, 189, 203, 243,
261, 273, 351, 377, 507, 567, 609, 783, 819, 1053, 1131, 1183, 1521,
1701, 1827, 2349, 2457, 2639, 3159, 3393, 3549, 4563, 4901, 5481, 7047,
7371, 7917, 10179, 10647, 13689, 14703, 16443, 22113, 23751, 30537, 31941,
34307, 41067, 44109, 49329, 71253, 91611, 95823, 102921,
132327, 213759, 287469, 308763, 396981, 641277, 926289,
1190943, 2778867, 8336601}, {1, 2, 3, 5, 6, 7, 9, 10, 14, 15,
18, 21, 27, 29, 30, 35, 42, 45, 49, 54, 58, 63, 70, 81, 87, 90,
98, 105, 126, 135, 145, 147, 162, 174, 189, 203,
210, 243, 245, 261, 270, 290, 294, 315, 378, 405, 406, 435, 441, 486, 490,
522, 567, 609, 630, 729, 735, 783, 810, 870, 882, 945, 1015,
1134, 1215, 1218, 1305, 1323, 1421, 1458, 1470, 1566, 1701, 1827, 1890,
2030, 2205, 2349, 2430, 2610, 2646, 2835, 2842, 3045, 3402, 3645,
3654, 3915, 3969, 4263, 4410, 4698, 5103, 5481, 5670, 6090, 6615, 7047,
7105, 7290, 7830, 7938, 8505, 8526, 9135, 10206, 10962, 11745, 11907,
12789, 13230, 14094, 14210, 16443, 17010, 18270, 19845, 21141, 21315,
23490, 23814, 25515, 25578, 27405, 32886, 35235, 35721, 38367, 39690,
42282, 42630, 49329, 51030, 54810, 59535, 63945, 70470,
71442, 76734, 82215, 98658, 105705, 115101, 119070, 127890, 147987,
164430, 178605, 191835, 211410, 230202, 246645, 295974, 345303, 357210,
383670, 493290, 575505, 690606, 739935, 1035909, 1151010, 1479870, 1726515,
2071818, 3453030, 5179545, 10359090}, {1, 2, 5, 7, 10, 14, 29, 31,
35, 49, 58, 62, 70, 98, 145, 155, 203, 217, 245, 290, 310, 406, 434,
490, 841, 899, 1015, 1085, 1421, 1519, 1682, 1798, 2030, 2170, 2842, 3038,
4205, 4495, 5887, 6293, 7105, 7595, 8410, 8990,
11774, 12586, 14210, 15190, 26071, 29435, 31465, 41209, 44051, 52142,
58870, 62930, 82418, 88102, 130355, 182497, 206045, 220255,
260710, 364994, 412090, 440510, 912485, 1277479, 1824970, 2554958,
6387395, 12774790}, {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 29, 30,
31, 40, 50, 58, 60, 62, 75, 87, 93, 100, 116, 120, 124, 145, 150, 155,
174, 186, 200, 232, 248, 290, 300, 310, 348, 372, 435, 465, 580,
600, 620, 696, 725, 744, 775, 841, 870, 899, 930, 1160, 1240, 1450,
1550, 1682, 1740, 1798, 1860, 2175, 2325, 2523, 2697, 2900, 3100, 3364,
3480, 3596, 3720, 4205, 4350, 4495, 4650, 5046, 5394, 5800,
6200, 6728, 7192, 8410,
8700, 8990, 9300, 10092, 10788, 12615, 13485, 16820, 17400, 17980, 18600,
20184, 21025, 21576, 22475, 25230, 26071, 26970, 33640, 35960, 42050, 44950,
50460, 52142, 53940, 63075, 67425, 78213, 84100, 89900, 100920,
104284, 107880, 126150, 130355, 134850, 156426, 168200, 179800, 208568,
252300, 260710, 269700, 312852, 391065, 504600, 521420, 539400, 625704,
651775, 782130, 1042840, 1303550, 1564260, 1955325, 2607100, 3128520,
3910650, 5214200, 7821300, 15642600}, {1, 2, 3, 4, 5, 6, 8, 9, 10,
11, 12, 15, 18, 20, 22, 24, 25, 30, 31, 33, 36, 40, 44, 45, 50,
55, 60, 62, 66, 72, 75, 88, 90, 93, 99, 100, 110, 120, 124, 132,
150, 155, 165, 180, 186, 198, 200, 220, 225, 248, 264, 275,
279, 300, 310, 330, 341, 360, 372, 396, 440, 450, 465, 495, 550, 558,
600, 620, 660, 682, 744, 775, 792, 825, 900, 930, 961, 990, 1023,
1100, 1116, 1240, 1320, 1364, 1395, 1550, 1650, 1705, 1800, 1860,
1922, 1980, 2046, 2200, 2232, 2325, 2475, 2728, 2790, 2883,
3069, 3100, 3300, 3410, 3720, 3844, 3960, 4092, 4650, 4805, 4950,
5115, 5580, 5766, 6138, 6200, 6600, 6820, 6975, 7688, 8184, 8525, 8649,
9300, 9610, 9900, 10230, 10571, 11160, 11532, 12276, 13640,
13950, 14415, 15345, 17050, 17298, 18600, 19220, 19800, 20460,
21142, 23064, 24025, 24552, 25575, 27900, 28830, 30690, 31713, 34100,
34596, 38440, 40920, 42284, 43245, 48050, 51150, 52855, 55800, 57660,
61380, 63426, 68200, 69192, 72075, 76725, 84568, 86490, 95139, 96100,
102300, 105710, 115320, 122760, 126852, 144150, 153450, 158565, 172980,
190278, 192200, 204600, 211420, 216225, 253704, 264275,
288300, 306900, 317130, 345960, 380556, 422840, 432450,
475695, 528550, 576600, 613800, 634260, 761112, 792825,
864900, 951390, 1057100, 1268520, 1585650, 1729800, 1902780, 2114200,
2378475, 3171300, 3805560, 4756950, 6342600, 9513900, 19027800}, {1, 2, 4,
8, 11, 16, 17, 22, 31, 32, 34, 44, 62, 64, 68, 88, 124, 128, 136, 176,
187, 248, 272, 341, 352, 374, 496, 527, 544, 682, 704, 748, 961, 992,
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krassi_holmz · 2006-03-05 06:33:45

Ansette wrote:

for n=3 i believe we've managed 1, but after that we're unsure

Who "we"?

Ansette · 2006-03-05 07:41:21

group of students for this work..

krassi_holmz · 2006-03-05 07:56:14

OK. Just don't forget putting my name in your presentation
I don't actually think I can handle with the 3rd problem.
Can you tell me more of that patterns?

Ansette · 2006-03-05 11:39:26

well so far we have (from n=1)

1,2,1,4,5..

but then we're not even certain these are right!

krassi_holmz · 2006-03-05 19:39:38

I have an idea.
I think I can prove your question number2 using the same methods as prob1:
Let

detones the sum of all products of k numbers for which:
.
We can write this function like this:

Understand?
Clearly

Last edited by krassi_holmz (2006-03-05 19:48:47)

krassi_holmz · 2006-03-05 19:41:14

So

krassi_holmz · 2006-03-05 20:08:46

OK. I'm done.
Here's the functional equation(I leave the froof to you):

So:

and, the main:

Math Is Fun Forum

#1 2006-03-05 03:40:44

Pure problem

#2 2006-03-05 04:19:12

Re: Pure problem

#3 2006-03-05 04:27:10

Re: Pure problem

#4 2006-03-05 04:31:05

Re: Pure problem

#5 2006-03-05 04:39:06

Re: Pure problem

#6 2006-03-05 04:46:11

Re: Pure problem

#7 2006-03-05 04:59:09

Re: Pure problem

#8 2006-03-05 05:07:49

Re: Pure problem

#9 2006-03-05 05:13:43

Re: Pure problem

#10 2006-03-05 05:16:23

Re: Pure problem

#11 2006-03-05 05:32:30

Re: Pure problem

#12 2006-03-05 05:39:26

Re: Pure problem

#13 2006-03-05 05:46:31

Re: Pure problem

#14 2006-03-05 06:04:47

Re: Pure problem

#15 2006-03-05 06:17:16

Re: Pure problem

#16 2006-03-05 06:18:55

Re: Pure problem

#17 2006-03-05 06:24:36

Re: Pure problem

#18 2006-03-05 06:26:01

Re: Pure problem

#19 2006-03-05 06:33:45

Re: Pure problem

#20 2006-03-05 07:41:21

Re: Pure problem

#21 2006-03-05 07:56:14

Re: Pure problem

#22 2006-03-05 11:39:26

Re: Pure problem

#23 2006-03-05 19:39:38

Re: Pure problem

#24 2006-03-05 19:41:14

Re: Pure problem

#25 2006-03-05 20:08:46

Re: Pure problem

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