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1 vote (5.88%)

17 members have voted

kubikulann
kubikulann
kubikulann
Joined: Jun 28, 2011
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August 21st, 2019 at 7:44:08 PM permalink
Seems like the columns tend to the Fibonacci series (1,1,2,3,5,8,13,21,34,55,…)
where one term is the sum of the two previous ones.
Reperiet qui quaesiverit
weezrDASvegas
weezrDASvegas
weezrDASvegas
Joined: Feb 2, 2018
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August 22nd, 2019 at 1:21:40 AM permalink
Quote: kubikulann

Quote: 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.
gordonm888
gordonm888
gordonm888
Joined: Feb 18, 2015
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Thanks for this post from:
Ajaxx
August 22nd, 2019 at 11:01:45 AM permalink
Quote: Wizard

Sorry 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!
Last edited by: gordonm888 on Aug 22, 2019
So many better men, a few of them friends, were dead. And a thousand thousand slimy things lived on, and so did I.
gordonm888
gordonm888
gordonm888
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August 23rd, 2019 at 6:02:10 AM permalink
The long silence after my last post leaves me wondering whether Wizard (and others) are attempting to write an algorithm, using combination math, to calculate the number of "pieces of 1" in partitions of any number N - as a way to calculate the partition number of any number N. I've been trying to conceptually develop the algorithm in my head, to judge whether its worth actually trying to attempt it. I don't know - its uncertain of success.

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.
Last edited by: gordonm888 on Aug 23, 2019
So many better men, a few of them friends, were dead. And a thousand thousand slimy things lived on, and so did I.
Wizard
Administrator
Wizard
Wizard
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August 25th, 2019 at 3:46:36 AM permalink
Quote: gordonm888

The 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.
It's not whether you win or lose; it's whether or not you had a good bet.
Wizard
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Wizard
Wizard
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August 25th, 2019 at 3:01:33 PM permalink
Let me put into a table, what you said, because I love tables. The third column represents the sum of partitions from 1 to n-1, for n. For examples, for n=5, it equals P(1)+ P(2) + P(3) + P(4). As Gordon said, this column equals one less than the number of pieces of size one in the partition of n.

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.
It's not whether you win or lose; it's whether or not you had a good bet.
Wizard
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Wizard
Wizard
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August 25th, 2019 at 3:35:07 PM permalink
I 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)

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.
It's not whether you win or lose; it's whether or not you had a good bet.
gordonm888
gordonm888
gordonm888
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August 25th, 2019 at 7:11:21 PM permalink
Quote: Wizard

I 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.
So many better men, a few of them friends, were dead. And a thousand thousand slimy things lived on, and so did I.
Wizard
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Wizard
Wizard
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August 25th, 2019 at 7:20:42 PM permalink
Quote: gordonm888

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.



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))
It's not whether you win or lose; it's whether or not you had a good bet.
gordonm888
gordonm888
gordonm888
Joined: Feb 18, 2015
  • Threads: 34
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August 25th, 2019 at 7:57:54 PM permalink
Quote: Wizard

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))



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.
So many better men, a few of them friends, were dead. And a thousand thousand slimy things lived on, and so did I.
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