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17 members have voted
where one term is the sum of the two previous ones.
Quote: kubikulannQuote: gordonm888
I note that one can assign a number of permutations to any partition (e.g.: the partition 3-2-1-1 has 12 permutations, ) and then sum up the number of permutations for every entry in this table. When I do that I get this table:
1 2 3 4 5 6 7 8 9 10 Total 1 1 1 2 1 1 2 3 1 2 1 4 4 1 3 3 1 8 5 1 4 6 4 1 16 6 1 5 10 10 5 1 32 7 1 6 15 20 15 6 1 64 8 1 7 21 35 35 21 7 1 128 9 1 8 28 56 70 56 28 8 1 256 10 1 9 36 84 126 126 84 36 9 1 512
This is Pascal’s table. (aka Binomial coefficients table)
Lottery numerology
A combo of 6 numbers occurs 2 times: diagonally and vertically
3 6 10 15 21 28
They are birthday numbers. Many people swear by birth numbers and some won millions with them in lottos all over the world.
Im gonna play em. Thanks all.
Quote: WizardSorry to interrupt a good discussion, but here is a count of the number of pieces by size. The left column shows the piece size and along the top row is the base number that is getting partitioned.
Piece 1 2 3 4 5 6 7 8 9 10 1 1 2 4 7 12 19 30 45 67 97 2 1 1 3 4 8 11 19 26 41 3 1 1 2 4 6 9 15 21 4 1 1 2 3 6 8 13 5 1 1 2 3 5 8 6 1 1 2 3 5 7 1 1 2 3 8 1 1 2 9 1 1 10 1 Total 1 3 6 12 20 35 54 86 128 192
Offhand, the only interesting pattern I'm seeing is the total number of pieces generally divide lots of ways:
6 = 2*3
12 = 2*2*3
20 =2*2*5
35 = 5*7
54 = 2*3*3*3
86 = 2 * 43
128 = 2*2*2*2*2*2*2
192 = 2*2*2*2*2*2*3
I guess 86 is the exception.
If you keep a running sum/total of the Partitions Numbers and add one to that running sum you get:
N...P(N)....(running Sum of P(N)) +1
1___1______2
2___2______4
3___3______7
4___5______12
5___7______19
6___11_____30
7___15_____45
8___22_____67
9___30_____97
Notice that the (running sum of the partition number +1) = number of pieces that are 1 !
Edit: Another way to say this: think of the "Pieces of 1" row as a sequence of numbers. If you take the nth term and subtract the (n-1) term, you obtain a sequence that is the partition numbers!!
When you look at primes, you can't find any patterns. When you look at partitions, everything is a recursive pattern! Everywhere you look!!! Its maddening!!!! It's TURTLES all the way down!
Edit: I think that such an attempt is likely to wind up with an algorithm that is a "sum of a sum" of terms, amounting to a sum of N[sup}2 terms. That's essentially the same as developing a table of N[sup}2 entries, as we are doing now. Unless there is a way to represent many terms by a function (like defining an infinite series to be a sin() function) I question whether such an approach will work.
Quote: gordonm888The long silence after my last post ...
Sorry I was out of town for a few days. I'll get caught up shortly. Just skimming this, I find it intriguing, but need to chew on it some more.
n | Partitions | Sum from 1 to n-1 | Size 1 |
---|---|---|---|
1 | 1 | 0 | 1 |
2 | 2 | 1 | 2 |
3 | 3 | 3 | 4 |
4 | 5 | 6 | 7 |
5 | 7 | 11 | 12 |
6 | 11 | 18 | 19 |
7 | 15 | 29 | 30 |
8 | 22 | 44 | 45 |
9 | 30 | 66 | 67 |
10 | 42 | 96 | 97 |
11 | 56 | 138 | 139 |
12 | 77 | 194 | 195 |
13 | 101 | 271 | 272 |
14 | 135 | 372 | 373 |
15 | 176 | 507 | 508 |
Offhand, I can't think of why this is true.
Here it is
P(n) = P(n-1) * exp(0.923266 * n-0.448858)
Total | Partitions | Estimate |
---|---|---|
2 | 2 | 2 |
3 | 3 | 4 |
4 | 5 | 5 |
5 | 7 | 8 |
6 | 11 | 11 |
7 | 15 | 16 |
8 | 22 | 22 |
9 | 30 | 31 |
10 | 42 | 42 |
11 | 56 | 58 |
12 | 77 | 76 |
13 | 101 | 103 |
14 | 135 | 134 |
15 | 176 | 178 |
16 | 231 | 230 |
17 | 297 | 299 |
18 | 385 | 382 |
19 | 490 | 492 |
20 | 627 | 623 |
21 | 792 | 793 |
22 | 1,002 | 997 |
23 | 1,255 | 1,256 |
24 | 1,575 | 1,567 |
25 | 1,958 | 1,958 |
26 | 2,436 | 2,425 |
27 | 3,010 | 3,006 |
28 | 3,718 | 3,702 |
29 | 4,565 | 4,558 |
30 | 5,604 | 5,579 |
31 | 6,842 | 6,829 |
32 | 8,349 | 8,314 |
33 | 10,143 | 10,118 |
34 | 12,310 | 12,261 |
35 | 14,883 | 14,844 |
36 | 17,977 | 17,904 |
37 | 21,637 | 21,578 |
38 | 26,015 | 25,914 |
39 | 31,185 | 31,093 |
40 | 37,338 | 37,197 |
41 | 44,583 | 44,450 |
42 | 53,174 | 52,975 |
43 | 63,261 | 63,069 |
44 | 75,175 | 74,902 |
45 | 89,134 | 88,857 |
46 | 105,558 | 105,184 |
47 | 124,754 | 124,368 |
48 | 147,273 | 146,758 |
49 | 173,525 | 172,989 |
50 | 204,226 | 203,529 |
51 | 239,943 | 239,201 |
52 | 281,589 | 280,650 |
53 | 329,931 | 328,922 |
54 | 386,155 | 384,890 |
55 | 451,276 | 449,910 |
56 | 526,823 | 525,136 |
57 | 614,154 | 612,313 |
58 | 715,220 | 712,982 |
59 | 831,820 | 829,364 |
60 | 966,467 | 963,500 |
61 | 1,121,505 | 1,118,247 |
62 | 1,300,156 | 1,296,257 |
63 | 1,505,499 | 1,501,189 |
64 | 1,741,630 | 1,736,523 |
65 | 2,012,558 | 2,006,902 |
66 | 2,323,520 | 2,316,851 |
67 | 2,679,689 | 2,672,297 |
68 | 3,087,735 | 3,079,074 |
69 | 3,554,345 | 3,544,718 |
70 | 4,087,968 | 4,076,762 |
71 | 4,697,205 | 4,684,738 |
72 | 5,392,783 | 5,378,324 |
73 | 6,185,689 | 6,169,606 |
74 | 7,089,500 | 7,070,934 |
75 | 8,118,264 | 8,097,582 |
76 | 9,289,091 | 9,265,325 |
77 | 10,619,863 | 10,593,389 |
78 | 12,132,164 | 12,101,831 |
79 | 13,848,650 | 13,814,875 |
80 | 15,796,476 | 15,757,907 |
81 | 18,004,327 | 17,961,369 |
82 | 20,506,255 | 20,457,367 |
83 | 23,338,469 | 23,284,033 |
84 | 26,543,660 | 26,481,857 |
85 | 30,167,357 | 30,098,590 |
86 | 34,262,962 | 34,185,095 |
87 | 38,887,673 | 38,801,042 |
88 | 44,108,109 | 44,010,271 |
89 | 49,995,925 | 49,887,144 |
90 | 56,634,173 | 56,511,563 |
91 | 64,112,359 | 63,976,143 |
92 | 72,533,807 | 72,380,601 |
93 | 82,010,177 | 81,840,049 |
94 | 92,669,720 | 92,478,779 |
95 | 104,651,419 | 104,439,541 |
96 | 118,114,304 | 117,876,907 |
97 | 133,230,930 | 132,967,732 |
98 | 150,198,136 | 149,903,764 |
99 | 169,229,875 | 168,903,715 |
100 | 190,569,292 | 190,205,143 |
101 | 214,481,126 | 214,077,983 |
102 | 241,265,379 | 240,815,950 |
103 | 271,248,950 | 270,751,831 |
104 | 304,801,365 | 304,248,019 |
105 | 342,325,709 | 341,714,098 |
106 | 384,276,336 | 383,596,583 |
107 | 431,149,389 | 430,398,690 |
108 | 483,502,844 | 482,669,611 |
109 | 541,946,240 | 541,026,864 |
110 | 607,163,746 | 606,144,673 |
111 | 679,903,203 | 678,779,658 |
112 | 761,002,156 | 759,758,436 |
113 | 851,376,628 | 850,006,583 |
114 | 952,050,665 | 950,535,914 |
115 | 1,064,144,451 | 1,062,477,315 |
116 | 1,188,908,248 | 1,187,067,276 |
117 | 1,327,710,076 | 1,325,685,559 |
118 | 1,482,074,143 | 1,479,841,222 |
119 | 1,653,668,665 | 1,651,215,227 |
120 | 1,844,349,560 | 1,841,646,583 |
121 | 2,056,148,051 | 2,053,180,746 |
122 | 2,291,320,912 | 2,288,055,476 |
123 | 2,552,338,241 | 2,548,756,438 |
124 | 2,841,940,500 | 2,838,003,205 |
125 | 3,163,127,352 | 3,158,812,305 |
126 | 3,519,222,692 | 3,514,484,355 |
127 | 3,913,864,295 | 3,908,675,862 |
128 | 4,351,078,600 | 4,345,387,189 |
129 | 4,835,271,870 | 4,829,045,048 |
130 | 5,371,315,400 | 5,364,492,060 |
131 | 5,964,539,504 | 5,957,080,654 |
132 | 6,620,830,889 | 6,612,665,660 |
133 | 7,346,629,512 | 7,337,711,517 |
134 | 8,149,040,695 | 8,139,287,915 |
135 | 9,035,836,076 | 9,025,193,137 |
136 | 10,015,581,680 | 10,003,954,084 |
137 | 11,097,645,016 | 11,084,967,033 |
138 | 12,292,341,831 | 12,278,504,303 |
139 | 13,610,949,895 | 13,595,875,397 |
140 | 15,065,878,135 | 15,049,440,854 |
141 | 16,670,689,208 | 16,652,797,825 |
142 | 18,440,293,320 | 18,420,803,387 |
143 | 20,390,982,757 | 20,369,787,021 |
144 | 22,540,654,445 | 22,517,586,846 |
145 | 24,908,858,009 | 24,883,793,414 |
146 | 27,517,052,599 | 27,489,800,523 |
147 | 30,388,671,978 | 30,359,086,343 |
148 | 33,549,419,497 | 33,517,282,606 |
149 | 37,027,355,200 | 36,992,497,240 |
150 | 40,853,235,313 | 40,815,407,361 |
151 | 45,060,624,582 | 45,019,630,316 |
152 | 49,686,288,421 | 49,641,843,916 |
153 | 54,770,336,324 | 54,722,214,741 |
154 | 60,356,673,280 | 60,304,551,933 |
155 | 66,493,182,097 | 66,436,799,884 |
156 | 73,232,243,759 | 73,171,234,091 |
157 | 80,630,964,769 | 80,565,028,699 |
158 | 88,751,778,802 | 88,680,500,984 |
159 | 97,662,728,555 | 97,585,767,147 |
160 | 107,438,159,466 | 107,355,045,666 |
161 | 118,159,068,427 | 118,069,412,967 |
162 | 129,913,904,637 | 129,817,179,016 |
163 | 142,798,995,930 | 142,694,759,153 |
164 | 156,919,475,295 | 156,807,133,501 |
165 | 172,389,800,255 | 172,268,855,018 |
166 | 189,334,822,579 | 189,204,609,387 |
167 | 207,890,420,102 | 207,750,378,180 |
168 | 228,204,732,751 | 228,054,120,182 |
169 | 250,438,925,115 | 250,277,114,645 |
170 | 274,768,617,130 | 274,594,783,747 |
171 | 301,384,802,048 | 301,198,246,569 |
172 | 330,495,499,613 | 330,295,309,281 |
173 | 362,326,859,895 | 362,112,259,825 |
174 | 397,125,074,750 | 396,895,058,709 |
175 | 435,157,697,830 | 434,911,413,024 |
176 | 476,715,857,290 | 476,452,200,458 |
177 | 522,115,831,195 | 521,833,869,016 |
178 | 571,701,605,655 | 571,400,136,521 |
179 | 625,846,753,120 | 625,524,763,617 |
180 | 684,957,390,936 | 684,613,579,960 |
181 | 749,474,411,781 | 749,107,689,772 |
182 | 819,876,908,323 | 819,485,877,701 |
183 | 896,684,817,527 | 896,268,317,743 |
184 | 980,462,880,430 | 980,019,425,239 |
185 | 1,071,823,774,337 | 1,071,352,143,426 |
186 | 1,171,432,692,373 | 1,170,931,321,289 |
187 | 1,280,011,042,268 | 1,279,478,668,917 |
188 | 1,398,341,745,571 | 1,397,776,745,817 |
189 | 1,527,273,599,625 | 1,526,674,692,637 |
190 | 1,667,727,404,093 | 1,667,092,933,883 |
191 | 1,820,701,100,652 | 1,820,029,801,774 |
192 | 1,987,276,856,363 | 1,986,567,076,583 |
193 | 2,168,627,105,469 | 2,167,877,641,991 |
194 | 2,366,022,741,845 | 2,365,231,996,716 |
195 | 2,580,840,212,973 | 2,580,007,105,134 |
196 | 2,814,570,987,591 | 2,813,694,041,537 |
197 | 3,068,829,878,530 | 3,067,908,214,547 |
198 | 3,345,365,983,698 | 3,344,398,333,926 |
199 | 3,646,072,432,125 | 3,645,058,221,777 |
200 | 3,972,999,029,388 | 3,971,937,310,627 |
201 | 4,328,363,658,647 | 4,327,254,289,804 |
202 | 4,714,566,886,083 | 4,713,409,383,776 |
203 | 5,134,205,287,973 | 5,133,000,108,179 |
204 | 5,590,088,317,495 | 5,588,835,619,535 |
205 | 6,085,253,859,260 | 6,083,954,904,602 |
206 | 6,622,987,708,040 | 6,621,643,524,989 |
207 | 7,206,841,706,490 | 7,205,454,616,281 |
208 | 7,840,656,226,137 | 7,839,228,409,394 |
209 | 8,528,581,302,375 | 8,527,116,458,389 |
210 | 9,275,102,575,355 | 9,273,604,382,976 |
211 | 10,085,065,885,767 | 10,083,539,816,166 |
212 | 10,963,707,205,259 | 10,962,158,865,096 |
213 | 11,916,681,236,278 | 11,915,118,346,825 |
214 | 12,950,095,925,895 | 12,948,526,550,206 |
215 | 14,070,545,699,287 | 14,068,980,396,734 |
216 | 15,285,151,248,481 | 15,283,601,178,668 |
217 | 16,601,598,107,914 | 16,600,077,386,512 |
218 | 18,028,182,516,671 | 18,026,706,182,776 |
219 | 19,573,856,161,145 | 19,572,442,752,352 |
220 | 21,248,279,009,367 | 21,246,948,396,225 |
221 | 23,061,871,173,849 | 23,060,647,367,795 |
222 | 25,025,873,760,111 | 25,024,782,607,963 |
223 | 27,152,408,925,615 | 27,151,481,184,166 |
224 | 29,454,549,941,750 | 29,453,818,819,391 |
225 | 31,946,390,696,157 | 31,945,895,218,868 |
226 | 34,643,126,322,519 | 34,642,908,729,066 |
227 | 37,561,133,582,570 | 37,561,243,002,581 |
228 | 40,718,063,627,362 | 40,718,553,325,339 |
229 | 44,132,934,884,255 | 44,133,866,294,224 |
230 | 47,826,239,745,920 | 47,827,679,552,474 |
231 | 51,820,051,838,712 | 51,822,076,403,342 |
232 | 56,138,148,670,947 | 56,140,840,962,281 |
233 | 60,806,135,438,329 | 60,809,589,881,049 |
234 | 65,851,585,970,275 | 65,855,905,223,838 |
235 | 71,304,185,514,919 | 71,309,485,804,774 |
236 | 77,195,892,663,512 | 77,202,300,401,270 |
237 | 83,561,103,925,871 | 83,568,761,572,559 |
238 | 90,436,839,668,817 | 90,445,902,217,271 |
239 | 97,862,933,703,585 | 97,873,575,120,469 |
240 | 105,882,246,722,733 | 105,894,656,301,136 |
241 | 114,540,884,553,038 | 114,555,274,014,365 |
242 | 123,888,443,077,259 | 123,905,042,795,369 |
243 | 133,978,259,344,888 | 133,997,326,179,546 |
244 | 144,867,692,496,445 | 144,889,505,925,148 |
245 | 156,618,412,527,946 | 156,643,283,280,695 |
246 | 169,296,722,391,554 | 169,324,988,512,763 |
247 | 182,973,889,854,026 | 183,005,926,250,793 |
248 | 197,726,516,681,672 | 197,762,731,133,712 |
249 | 213,636,919,820,625 | 213,677,763,545,749 |
250 | 230,793,554,364,681 | 230,839,518,036,227 |
251 | 249,291,451,168,559 | 249,343,076,603,055 |
252 | 269,232,701,252,579 | 269,290,577,482,513 |
253 | 290,726,957,916,112 | 290,791,734,156,514 |
254 | 313,891,991,306,665 | 313,964,373,128,410 |
255 | 338,854,264,248,680 | 338,935,027,975,330 |
256 | 365,749,566,870,782 | 365,839,555,950,559 |
257 | 394,723,676,655,357 | 394,823,817,645,196 |
258 | 425,933,084,409,356 | 426,044,383,633,593 |
259 | 459,545,750,448,675 | 459,669,311,792,618 |
260 | 495,741,934,760,846 | 495,878,956,707,614 |
261 | 534,715,062,908,609 | 534,866,858,363,775 |
262 | 576,672,674,947,168 | 576,840,668,810,756 |
263 | 621,837,416,509,615 | 622,023,167,767,158 |
264 | 670,448,123,060,170 | 670,653,323,043,735 |
265 | 722,760,953,690,372 | 722,987,450,753,395 |
266 | 779,050,629,562,167 | 779,300,428,193,871 |
267 | 839,611,730,366,814 | 839,887,018,776,921 |
268 | 904,760,108,316,360 | 905,063,258,652,871 |
269 | 974,834,369,944,625 | 975,167,969,139,993 |
270 | 1,050,197,489,931,110 | 1,050,564,341,278,630 |
271 | 1,131,238,503,938,600 | 1,131,641,661,657,750 |
272 | 1,218,374,349,844,330 | 1,218,817,122,296,590 |
273 | 1,312,051,800,816,210 | 1,312,537,789,269,910 |
274 | 1,412,749,565,173,450 | 1,413,282,669,049,830 |
275 | 1,520,980,492,851,170 | 1,521,564,953,213,890 |
276 | 1,637,293,969,337,170 | 1,637,934,376,582,770 |
277 | 1,762,278,433,057,260 | 1,762,979,775,792,390 |
278 | 1,896,564,103,591,580 | 1,897,331,779,227,310 |
279 | 2,040,825,852,575,070 | 2,041,665,722,301,010 |
280 | 2,195,786,311,682,510 | 2,196,704,714,572,330 |
281 | 2,362,219,145,337,710 | 2,363,222,960,201,970 |
282 | 2,540,952,590,045,690 | 2,542,049,253,703,810 |
283 | 2,732,873,183,547,530 | 2,734,070,760,525,010 |
284 | 2,938,929,793,929,550 | 2,940,236,999,631,600 |
285 | 3,160,137,867,148,990 | 3,161,564,146,448,330 |
286 | 3,397,584,011,986,770 | 3,399,139,568,230,940 |
287 | 3,652,430,836,071,050 | 3,654,126,719,723,870 |
288 | 3,925,922,161,489,420 | 3,927,770,305,994,760 |
289 | 4,219,388,528,587,090 | 4,221,401,850,472,780 |
290 | 4,534,253,126,900,880 | 4,536,445,569,650,230 |
291 | 4,872,038,056,472,080 | 4,874,424,703,634,060 |
292 | 5,234,371,069,753,670 | 5,236,968,198,243,810 |
293 | 5,622,992,691,950,600 | 5,625,817,899,891,690 |
294 | 6,039,763,882,095,510 | 6,042,836,153,121,440 |
295 | 6,486,674,127,079,080 | 6,490,013,974,965,660 |
296 | 6,965,850,144,195,830 | 6,969,479,689,946,320 |
297 | 7,479,565,078,510,580 | 7,483,508,214,051,230 |
298 | 8,030,248,384,943,040 | 8,034,530,865,140,210 |
299 | 8,620,496,275,465,020 | 8,625,145,903,435,480 |
300 | 9,253,082,936,723,600 | 9,258,129,673,182,110 |
301 | 9,930,972,392,403,500 | 9,936,448,565,599,450 |
302 | 10,657,331,232,548,800 | 10,663,271,667,656,800 |
303 | 11,435,542,077,822,100 | 11,441,984,334,851,100 |
304 | 12,269,218,019,229,400 | 12,276,202,545,689,500 |
305 | 13,162,217,895,057,700 | 13,169,788,295,603,300 |
306 | 14,118,662,665,280,000 | 14,126,865,881,272,200 |
307 | 15,142,952,738,857,100 | 15,151,839,354,143,200 |
308 | 16,239,786,535,829,600 | 16,249,410,987,302,400 |
309 | 17,414,180,133,147,200 | 17,424,601,057,574,000 |
310 | 18,671,488,299,600,300 | 18,682,768,780,005,900 |
311 | 20,017,426,762,576,900 | 20,029,634,721,655,400 |
312 | 21,458,096,037,352,800 | 21,471,304,525,099,900 |
313 | 23,000,006,655,487,300 | 23,014,294,295,620,400 |
314 | 24,650,106,150,830,400 | 24,665,557,475,818,700 |
315 | 26,415,807,633,566,300 | 26,432,513,591,256,500 |
316 | 28,305,020,340,996,000 | 28,323,078,684,208,600 |
317 | 30,326,181,989,842,900 | 30,345,697,851,320,000 |
318 | 32,488,293,351,466,600 | 32,509,379,696,109,400 |
319 | 34,800,954,869,440,800 | 34,823,733,146,946,400 |
320 | 37,274,405,776,748,000 | 37,299,006,445,639,900 |
321 | 39,919,565,526,999,900 | 39,946,128,795,469,600 |
322 | 42,748,078,035,954,600 | 42,776,754,468,280,500 |
323 | 45,772,358,543,578,000 | 45,803,309,901,036,700 |
324 | 49,005,643,635,237,800 | 49,039,043,576,859,200 |
325 | 52,462,044,228,828,600 | 52,498,079,266,012,700 |
326 | 56,156,602,112,874,200 | 56,195,472,418,011,800 |
327 | 60,105,349,839,666,500 | 60,147,270,329,785,500 |
328 | 64,325,374,609,114,500 | 64,370,575,877,957,000 |
329 | 68,834,885,946,073,800 | 68,883,615,494,065,300 |
330 | 73,653,287,861,850,300 | 73,705,811,169,211,700 |
331 | 78,801,255,302,666,600 | 78,857,857,225,508,800 |
332 | 84,300,815,636,225,100 | 84,361,801,640,598,500 |
333 | 90,175,434,980,549,600 | 90,241,132,726,932,600 |
334 | 96,450,110,192,202,700 | 96,520,870,953,318,000 |
335 | 103,151,466,321,735,000 | 103,227,666,780,563,000 |
336 | 110,307,860,425,292,000 | 110,389,904,302,398,000 |
337 | 117,949,491,546,113,000 | 118,037,811,639,875,000 |
338 | 126,108,517,833,796,000 | 126,203,577,886,370,000 |
339 | 134,819,180,623,301,000 | 134,921,477,635,335,000 |
340 | 144,117,936,527,873,000 | 144,228,002,896,388,000 |
341 | 154,043,597,379,576,000 | 154,162,003,523,593,000 |
342 | 164,637,479,165,761,000 | 164,764,835,973,643,000 |
343 | 175,943,559,810,422,000 | 176,080,521,617,844,000 |
344 | 188,008,647,052,292,000 | 188,155,914,441,466,000 |
345 | 200,882,556,287,683,000 | 201,040,879,464,350,000 |
346 | 214,618,299,743,286,000 | 214,788,481,736,150,000 |
347 | 229,272,286,871,217,000 | 229,455,187,360,548,000 |
348 | 244,904,537,455,382,000 | 245,101,076,427,177,000 |
349 | 261,578,907,351,144,000 | 261,790,069,437,081,000 |
350 | 279,363,328,483,702,000 | 279,590,167,131,561,000 |
351 | 298,330,063,062,758,000 | 298,573,705,454,962,000 |
352 | 318,555,973,788,329,000 | 318,817,625,598,558,000 |
353 | 340,122,810,048,577,000 | 340,403,761,014,690,000 |
354 | 363,117,512,048,110,000 | 363,419,141,393,747,000 |
355 | 387,632,532,919,029,000 | 387,956,315,666,778,000 |
356 | 413,766,180,933,342,000 | 414,113,694,080,277,000 |
357 | 441,622,981,929,358,000 | 441,995,911,597,065,000 |
358 | 471,314,064,268,398,000 | 471,714,212,733,254,000 |
359 | 502,957,566,506,000,000 | 503,386,860,294,917,000 |
360 | 536,679,070,310,691,000 | 537,139,568,199,682,000 |
361 | 572,612,058,898,037,000 | 573,105,961,076,774,000 |
362 | 610,898,403,751,884,000 | 611,428,060,918,555,000 |
363 | 651,688,879,997,206,000 | 652,256,803,730,406,000 |
364 | 695,143,713,458,946,000 | 695,752,586,553,637,000 |
365 | 741,433,159,884,081,000 | 742,085,848,086,667,000 |
366 | 790,738,119,649,411,000 | 791,437,683,397,883,000 |
367 | 843,250,788,562,528,000 | 844,000,496,260,715,000 |
368 | 899,175,348,396,088,000 | 899,978,689,741,625,000 |
369 | 958,728,697,912,338,000 | 959,589,398,908,019,000 |
370 | 1,022,141,228,367,340,000 | 1,023,063,266,444,290,000 |
371 | 1,089,657,644,424,390,000 | 1,090,645,265,413,150,000 |
372 | 1,161,537,834,849,960,000 | 1,162,595,570,132,140,000 |
373 | 1,238,057,794,119,120,000 | 1,239,190,479,809,280,000 |
374 | 1,319,510,599,727,470,000 | 1,320,723,396,115,790,000 |
375 | 1,406,207,446,561,480,000 | 1,407,505,859,788,460,000 |
376 | 1,498,478,743,590,580,000 | 1,499,868,647,676,860,000 |
377 | 1,596,675,274,490,750,000 | 1,598,162,935,821,660,000 |
378 | 1,701,169,427,975,810,000 | 1,702,761,530,251,060,000 |
379 | 1,812,356,499,739,470,000 | 1,814,060,171,624,690,000 |
380 | 1,930,656,072,350,460,000 | 1,932,478,915,720,990,000 |
381 | 2,056,513,475,336,630,000 | 2,058,463,596,496,840,000 |
382 | 2,190,401,332,423,760,000 | 2,192,487,374,066,330,000 |
383 | 2,332,821,198,543,890,000 | 2,335,052,374,987,350,000 |
384 | 2,484,305,294,265,410,000 | 2,486,691,427,602,110,000 |
385 | 2,645,418,340,688,760,000 | 2,647,969,900,547,130,000 |
386 | 2,816,759,503,217,940,000 | 2,819,487,647,630,880,000 |
387 | 2,998,964,447,736,450,000 | 3,001,881,067,996,670,000 |
388 | 3,192,707,518,433,530,000 | 3,195,825,285,280,280,000 |
389 | 3,398,704,041,358,160,000 | 3,402,036,455,563,610,000 |
390 | 3,617,712,763,867,600,000 | 3,621,274,208,413,110,000 |
391 | 3,850,538,434,667,420,000 | 3,854,344,231,777,640,000 |
392 | 4,098,034,535,626,590,000 | 4,102,101,005,688,210,000 |
393 | 4,361,106,170,762,280,000 | 4,365,450,696,609,950,000 |
394 | 4,640,713,124,699,620,000 | 4,645,354,218,123,350,000 |
395 | 4,937,873,096,788,190,000 | 4,942,830,470,972,180,000 |
396 | 5,253,665,124,416,970,000 | 5,258,959,768,984,820,000 |
397 | 5,589,233,202,595,400,000 | 5,594,887,465,213,220,000 |
398 | 5,945,790,114,707,870,000 | 5,951,827,785,731,630,000 |
399 | 6,324,621,482,504,290,000 | 6,331,067,886,881,070,000 |
400 | 6,727,090,051,741,040,000 | 6,733,972,144,452,060,000 |
401 | 7,154,640,222,653,940,000 | 7,161,986,692,184,490,000 |
402 | 7,608,802,843,339,870,000 | 7,616,644,219,256,550,000 |
403 | 8,091,200,276,484,460,000 | 8,099,569,045,897,860,000 |
I'm sure the estimate would become more accurate if I threw in more decimal places.
Quote: WizardI found a very good estimate for the P(n) if you know P(n-1).
Here it is
P(n) = P(n-1) * exp(0.923266 * n-0.448858)
{snip}
I'm sure the estimate would become more accurate if I threw in more decimal places.
I think you could come up with a more accurate version of this formula if you derived two different formulas: one each for even and odd values of n. If you use one formula, you should always tend to overestimate for odd n and underestimate for even n.
Quote: gordonm888I think you could come up with a more accurate version of this formula if you derived two different formulas: one each for even and odd values of n. If you use one formula, you should always tend to overestimate for odd n and underestimate for even n.
Doesn't this odd and even effect diminish over time? I did some calculus to get at a direct formula. However, there is already a good formula for this. I'll put it in spoiler tags. I'll just say that it goes back to what I wrote earlier, that series seems to always come down to pi, e, and primes. I'll add to that the Fibonacci series.
P(n) = 1/(4n*sqrt(3)) * exp(pi * sqrt(2n/3))
Quote: WizardDoesn't this odd and even effect diminish over time? I did some calculus to get at a direct formula. However, there is already a good formula for this. I'll put it in spoiler tags. I'll just say that it goes back to what I wrote earlier, that series seems to always come down to pi, e, and primes. I'll add to that the Fibonacci series.
P(n) = 1/(4n*sqrt(3)) * exp(pi * sqrt(2n/3))
I've looked at this only up from n=1 . . . 51. The odd/even effect definitely appears to get larger on an absolute basis but smaller on a % basis. At n=50,51 the P(n) for odd and even ns may be off by +/- 75, but the P(n) is >200,000 so as a % it is a small effect.