[seqfan] Re: Partition numbers and Fibonacci numbers
Susanne Wienand
susanne.wienand at gmail.com
Wed Oct 17 14:05:45 CEST 2012
A table (in the attachment) with the values of S, J, j, T... during the
iteration for m = 2 (this is m = 0 in the number triangle) also seems to
show a connection to the fibonacci sequence. The first two values of T in
an iteration at a particular n are the iteration-results for n-1 and n-2.
The values of T seem to be added up and subtracted in a certain way to get
the result for n.
Regards,
Susanne
2012/10/14 Jonathan Stauduhar <jstdhr at gmail.com>
> I think this might help?
>
> http://www.youtube.com/watch?**v=5TVOkD7hvCY<http://www.youtube.com/watch?v=5TVOkD7hvCY>
>
>
> On 10/13/2012 2:22 PM, Peter Luschny wrote:
>
>> Today I was looking at this number triangle, but I could
>> not make sense of it.
>>
>> 0: [ 1]
>> 1: [ 1, 1]
>> 2: [ 2, 2, 2]
>> 3: [ 3, 3, 3, 3]
>> 4: [ 5, 5, 5, 5, 5]
>> 5: [ 7, 8, 8, 8, 8, 8]
>> 6: [11, 14, 13, 13, 13, 13, 13]
>> 7: [15, 23, 22, 21, 21, 21, 21, 21]
>> 8: [22, 39, 36, 35, 34, 34, 34, 34, 34]
>> 9: [30, 65, 60, 57, 56, 55, 55, 55, 55, 55]
>> 10: [42, 109, 99, 94, 91, 90, 89, 89, 89, 89, 89]
>> 11: [56, 183, 164, 154, 149, 146, 145, 144, 144, 144, 144, 144]
>>
>> In the first column are the partition numbers and on the
>> diagonal are the Fibonacci numbers. Can anybody give
>> or point to an interpretation of this connection?
>>
>> The formal definition in `Sage´ is:
>>
>> @CachedFunction
>> def PartToFibo(n, m):
>> if n< 2: return 1
>> S = 0; J = n-1; j = m
>> while 0< J:
>> T = PartToFibo(J, m)
>> S = S-T if (j//m)%m == 0 else S+T
>> J -= j//m if j%m == 0 else j
>> j += 1
>> return S
>>
>> for n in (0..12): [PartToFibo(n+1,m+2) for m in (0..n)]
>>
>> Peter
>>
>> http://oeis.org/A000041 http://oeis.org/A000045
>>
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-------------- next part --------------
n S J j T j/m j%m (j/m)%m S J j result of iteration
-------------------------------------------------------------------------------------------------------------------------------------------
1 1
2 0 1 2 1 1 0 1 1 0 3
2 1 0 3 1
3 0 2 2 1 1 0 1 1 1 3
3 1 1 3 1 1 1 1 2 -2 4
3 2 -2 4 2
4 0 3 2 2 1 0 1 2 2 3
4 2 2 3 1 1 1 1 3 -1 4
4 3 -1 4 3
5 0 4 2 3 1 0 1 3 3 3
5 3 3 3 2 1 1 1 5 0 4
5 5 0 4 5
6 0 5 2 5 1 0 1 5 4 3
6 5 4 3 3 1 1 1 8 1 4
6 8 1 4 1 2 0 0 7 -1 5
6 7 -1 5 7
7 0 6 2 7 1 0 1 7 5 3
7 7 5 3 5 1 1 1 12 2 4
7 12 2 4 1 2 0 0 11 0 5
7 11 0 5 11
8 0 7 2 11 1 0 1 11 6 3
8 11 6 3 7 1 1 1 18 3 4
8 18 3 4 2 2 0 0 16 1 5
8 16 1 5 1 2 1 0 15 -4 6
8 15 -4 6 15
9 0 8 2 15 1 0 1 15 7 3
9 15 7 3 11 1 1 1 26 4 4
9 26 4 4 3 2 0 0 23 2 5
9 23 2 5 1 2 1 0 22 -3 6
9 22 -3 6 22
10 0 9 2 22 1 0 1 22 8 3
10 22 8 3 15 1 1 1 37 5 4
10 37 5 4 5 2 0 0 32 3 5
10 32 3 5 2 2 1 0 30 -2 6
10 30 -2 6 30
11 0 10 2 30 1 0 1 30 9 3
11 30 9 3 22 1 1 1 52 6 4
11 52 6 4 7 2 0 0 45 4 5
11 45 4 5 3 2 1 0 42 -1 6
11 42 -1 6 42
12 0 11 2 42 1 0 1 42 10 3
12 42 10 3 30 1 1 1 72 7 4
12 72 7 4 11 2 0 0 61 5 5
12 61 5 5 5 2 1 0 56 0 6
12 56 0 6 56
13 0 12 2 56 1 0 1 56 11 3
13 56 11 3 42 1 1 1 98 8 4
13 98 8 4 15 2 0 0 83 6 5
13 83 6 5 7 2 1 0 76 1 6
13 76 1 6 1 3 0 1 77 -2 7
13 77 -2 7 77
14 0 13 2 77 1 0 1 77 12 3
14 77 12 3 56 1 1 1 133 9 4
14 133 9 4 22 2 0 0 111 7 5
14 111 7 5 11 2 1 0 100 2 6
14 100 2 6 1 3 0 1 101 -1 7
14 101 -1 7 101
15 0 14 2 101 1 0 1 101 13 3
15 101 13 3 77 1 1 1 178 10 4
15 178 10 4 30 2 0 0 148 8 5
15 148 8 5 15 2 1 0 133 3 6
15 133 3 6 2 3 0 1 135 0 7
15 135 0 7 135
16 0 15 2 135 1 0 1 135 14 3
16 135 14 3 101 1 1 1 236 11 4
16 236 11 4 42 2 0 0 194 9 5
16 194 9 5 22 2 1 0 172 4 6
16 172 4 6 3 3 0 1 175 1 7
16 175 1 7 1 3 1 1 176 -6 8
16 176 -6 8 176
17 0 16 2 176 1 0 1 176 15 3
17 176 15 3 135 1 1 1 311 12 4
17 311 12 4 56 2 0 0 255 10 5
17 255 10 5 30 2 1 0 225 5 6
17 225 5 6 5 3 0 1 230 2 7
17 230 2 7 1 3 1 1 231 -5 8
17 231 -5 8 231
18 0 17 2 231 1 0 1 231 16 3
18 231 16 3 176 1 1 1 407 13 4
18 407 13 4 77 2 0 0 330 11 5
18 330 11 5 42 2 1 0 288 6 6
18 288 6 6 7 3 0 1 295 3 7
18 295 3 7 2 3 1 1 297 -4 8
18 297 -4 8 297
19 0 18 2 297 1 0 1 297 17 3
19 297 17 3 231 1 1 1 528 14 4
19 528 14 4 101 2 0 0 427 12 5
19 427 12 5 56 2 1 0 371 7 6
19 371 7 6 11 3 0 1 382 4 7
19 382 4 7 3 3 1 1 385 -3 8
19 385 -3 8 385
20 0 19 2 385 1 0 1 385 18 3
20 385 18 3 297 1 1 1 682 15 4
20 682 15 4 135 2 0 0 547 13 5
20 547 13 5 77 2 1 0 470 8 6
20 470 8 6 15 3 0 1 485 5 7
20 485 5 7 5 3 1 1 490 -2 8
20 490 -2 8 490
21 0 20 2 490 1 0 1 490 19 3
21 490 19 3 385 1 1 1 875 16 4
21 875 16 4 176 2 0 0 699 14 5
21 699 14 5 101 2 1 0 598 9 6
21 598 9 6 22 3 0 1 620 6 7
21 620 6 7 7 3 1 1 627 -1 8
21 627 -1 8 627
22 0 21 2 627 1 0 1 627 20 3
22 627 20 3 490 1 1 1 1117 17 4
22 1117 17 4 231 2 0 0 886 15 5
22 886 15 5 135 2 1 0 751 10 6
22 751 10 6 30 3 0 1 781 7 7
22 781 7 7 11 3 1 1 792 0 8
22 792 0 8 792
23 0 22 2 792 1 0 1 792 21 3
23 792 21 3 627 1 1 1 1419 18 4
23 1419 18 4 297 2 0 0 1122 16 5
23 1122 16 5 176 2 1 0 946 11 6
23 946 11 6 42 3 0 1 988 8 7
23 988 8 7 15 3 1 1 1003 1 8
23 1003 1 8 1 4 0 0 1002 -3 9
23 1002 -3 9 1002
24 0 23 2 1002 1 0 1 1002 22 3
24 1002 22 3 792 1 1 1 1794 19 4
24 1794 19 4 385 2 0 0 1409 17 5
24 1409 17 5 231 2 1 0 1178 12 6
24 1178 12 6 56 3 0 1 1234 9 7
24 1234 9 7 22 3 1 1 1256 2 8
24 1256 2 8 1 4 0 0 1255 -2 9
24 1255 -2 9 1255
25 0 24 2 1255 1 0 1 1255 23 3
25 1255 23 3 1002 1 1 1 2257 20 4
25 2257 20 4 490 2 0 0 1767 18 5
25 1767 18 5 297 2 1 0 1470 13 6
25 1470 13 6 77 3 0 1 1547 10 7
25 1547 10 7 30 3 1 1 1577 3 8
25 1577 3 8 2 4 0 0 1575 -1 9
25 1575 -1 9 1575
26 0 25 2 1575 1 0 1 1575 24 3
26 1575 24 3 1255 1 1 1 2830 21 4
26 2830 21 4 627 2 0 0 2203 19 5
26 2203 19 5 385 2 1 0 1818 14 6
26 1818 14 6 101 3 0 1 1919 11 7
26 1919 11 7 42 3 1 1 1961 4 8
26 1961 4 8 3 4 0 0 1958 0 9
26 1958 0 9 1958
27 0 26 2 1958 1 0 1 1958 25 3
27 1958 25 3 1575 1 1 1 3533 22 4
27 3533 22 4 792 2 0 0 2741 20 5
27 2741 20 5 490 2 1 0 2251 15 6
27 2251 15 6 135 3 0 1 2386 12 7
27 2386 12 7 56 3 1 1 2442 5 8
27 2442 5 8 5 4 0 0 2437 1 9
27 2437 1 9 1 4 1 0 2436 -8 10
27 2436 -8 10 2436
28 0 27 2 2436 1 0 1 2436 26 3
28 2436 26 3 1958 1 1 1 4394 23 4
28 4394 23 4 1002 2 0 0 3392 21 5
28 3392 21 5 627 2 1 0 2765 16 6
28 2765 16 6 176 3 0 1 2941 13 7
28 2941 13 7 77 3 1 1 3018 6 8
28 3018 6 8 7 4 0 0 3011 2 9
28 3011 2 9 1 4 1 0 3010 -7 10
28 3010 -7 10 3010
29 0 28 2 3010 1 0 1 3010 27 3
29 3010 27 3 2436 1 1 1 5446 24 4
29 5446 24 4 1255 2 0 0 4191 22 5
29 4191 22 5 792 2 1 0 3399 17 6
29 3399 17 6 231 3 0 1 3630 14 7
29 3630 14 7 101 3 1 1 3731 7 8
29 3731 7 8 11 4 0 0 3720 3 9
29 3720 3 9 2 4 1 0 3718 -6 10
29 3718 -6 10 3718
30 0 29 2 3718 1 0 1 3718 28 3
30 3718 28 3 3010 1 1 1 6728 25 4
30 6728 25 4 1575 2 0 0 5153 23 5
30 5153 23 5 1002 2 1 0 4151 18 6
30 4151 18 6 297 3 0 1 4448 15 7
30 4448 15 7 135 3 1 1 4583 8 8
30 4583 8 8 15 4 0 0 4568 4 9
30 4568 4 9 3 4 1 0 4565 -5 10
30 4565 -5 10 4565
31 0 30 2 4565 1 0 1 4565 29 3
31 4565 29 3 3718 1 1 1 8283 26 4
31 8283 26 4 1958 2 0 0 6325 24 5
31 6325 24 5 1255 2 1 0 5070 19 6
31 5070 19 6 385 3 0 1 5455 16 7
31 5455 16 7 176 3 1 1 5631 9 8
31 5631 9 8 22 4 0 0 5609 5 9
31 5609 5 9 5 4 1 0 5604 -4 10
31 5604 -4 10 5604
32 0 31 2 5604 1 0 1 5604 30 3
32 5604 30 3 4565 1 1 1 10169 27 4
32 10169 27 4 2436 2 0 0 7733 25 5
32 7733 25 5 1575 2 1 0 6158 20 6
32 6158 20 6 490 3 0 1 6648 17 7
32 6648 17 7 231 3 1 1 6879 10 8
32 6879 10 8 30 4 0 0 6849 6 9
32 6849 6 9 7 4 1 0 6842 -3 10
32 6842 -3 10 6842
33 0 32 2 6842 1 0 1 6842 31 3
33 6842 31 3 5604 1 1 1 12446 28 4
33 12446 28 4 3010 2 0 0 9436 26 5
33 9436 26 5 1958 2 1 0 7478 21 6
33 7478 21 6 627 3 0 1 8105 18 7
33 8105 18 7 297 3 1 1 8402 11 8
33 8402 11 8 42 4 0 0 8360 7 9
33 8360 7 9 11 4 1 0 8349 -2 10
33 8349 -2 10 8349
34 0 33 2 8349 1 0 1 8349 32 3
34 8349 32 3 6842 1 1 1 15191 29 4
34 15191 29 4 3718 2 0 0 11473 27 5
34 11473 27 5 2436 2 1 0 9037 22 6
34 9037 22 6 792 3 0 1 9829 19 7
34 9829 19 7 385 3 1 1 10214 12 8
34 10214 12 8 56 4 0 0 10158 8 9
34 10158 8 9 15 4 1 0 10143 -1 10
34 10143 -1 10 10143
35 0 34 2 10143 1 0 1 10143 33 3
35 10143 33 3 8349 1 1 1 18492 30 4
35 18492 30 4 4565 2 0 0 13927 28 5
35 13927 28 5 3010 2 1 0 10917 23 6
35 10917 23 6 1002 3 0 1 11919 20 7
35 11919 20 7 490 3 1 1 12409 13 8
35 12409 13 8 77 4 0 0 12332 9 9
35 12332 9 9 22 4 1 0 12310 0 10
35 12310 0 10 12310
36 0 35 2 12310 1 0 1 12310 34 3
36 12310 34 3 10143 1 1 1 22453 31 4
36 22453 31 4 5604 2 0 0 16849 29 5
36 16849 29 5 3718 2 1 0 13131 24 6
36 13131 24 6 1255 3 0 1 14386 21 7
36 14386 21 7 627 3 1 1 15013 14 8
36 15013 14 8 101 4 0 0 14912 10 9
36 14912 10 9 30 4 1 0 14882 1 10
36 14882 1 10 1 5 0 1 14883 -4 11
36 14883 -4 11 14883
37 0 36 2 14883 1 0 1 14883 35 3
37 14883 35 3 12310 1 1 1 27193 32 4
37 27193 32 4 6842 2 0 0 20351 30 5
37 20351 30 5 4565 2 1 0 15786 25 6
37 15786 25 6 1575 3 0 1 17361 22 7
37 17361 22 7 792 3 1 1 18153 15 8
37 18153 15 8 135 4 0 0 18018 11 9
37 18018 11 9 42 4 1 0 17976 2 10
37 17976 2 10 1 5 0 1 17977 -3 11
37 17977 -3 11 17977
38 0 37 2 17977 1 0 1 17977 36 3
38 17977 36 3 14883 1 1 1 32860 33 4
38 32860 33 4 8349 2 0 0 24511 31 5
38 24511 31 5 5604 2 1 0 18907 26 6
38 18907 26 6 1958 3 0 1 20865 23 7
38 20865 23 7 1002 3 1 1 21867 16 8
38 21867 16 8 176 4 0 0 21691 12 9
38 21691 12 9 56 4 1 0 21635 3 10
38 21635 3 10 2 5 0 1 21637 -2 11
38 21637 -2 11 21637
39 0 38 2 21637 1 0 1 21637 37 3
39 21637 37 3 17977 1 1 1 39614 34 4
39 39614 34 4 10143 2 0 0 29471 32 5
39 29471 32 5 6842 2 1 0 22629 27 6
39 22629 27 6 2436 3 0 1 25065 24 7
39 25065 24 7 1255 3 1 1 26320 17 8
39 26320 17 8 231 4 0 0 26089 13 9
39 26089 13 9 77 4 1 0 26012 4 10
39 26012 4 10 3 5 0 1 26015 -1 11
39 26015 -1 11 26015
40 0 39 2 26015 1 0 1 26015 38 3
40 26015 38 3 21637 1 1 1 47652 35 4
40 47652 35 4 12310 2 0 0 35342 33 5
40 35342 33 5 8349 2 1 0 26993 28 6
40 26993 28 6 3010 3 0 1 30003 25 7
40 30003 25 7 1575 3 1 1 31578 18 8
40 31578 18 8 297 4 0 0 31281 14 9
40 31281 14 9 101 4 1 0 31180 5 10
40 31180 5 10 5 5 0 1 31185 0 11
40 31185 0 11 31185
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