|Jun7-11, 05:49 AM||#1|
Blog Entries: 2
ENA molecular structure related to continued fractions
I discovered a link between my series formulation, see below, and RNA molecule structure, In particular the RNA sequences A004148,A089735,A098075, A091964 and A124428 as posted in the Online Encyclopedia of Integer Sequences are directly related. See in particular http://oeis.org/A004148 and http://oeis.org/A091964
The relation has to do with the following triangle
01 02 01
02 03 03 01
04 06 06 04 01
08 13 13 10 05 01
17 28 30 24 15 06 01
37 62 69 59 40 21 07 01
The 1st column is sequence A004148 (verified to 12 terms)
The 2nd column is sequence A089735 (verified to 11 terms)
The 3rd column is sequence A098075 (verified to 10 terms)
The row sums 1,2,4,9,21,50,121,296 ... is sequence A091964 verified to 12 terms. The same row sums are the diagonal sums of the triangle A124428. Each of these sequences are related to RNA molecule structure as noted in the comments about these sequences and the sequences reference each other. Yet before I knew about these sequences, I formulated the above triangle based solely on my post regarding continued fractions and triangular numbers. See "Certain series related to continued fractions and triangular numbers" http://tech.groups.yahoo.com/group/N...up/message/145 In particular, if each row (from i = 1 for the beginning of the row to i = n at the end of the row) is taken as coefficients respectively of the 2i th term of my modified continued fraction sequence for a given m, then the sum adds up to the 2n th term of the corresponding sequence where m is increased by 1. Using the 4th row as an example, amd the "m" sequences as an example: For m = 1 the sequence is 0,1,.1.,2,.3.,5,.8.,13,.21,; the m = 2 sequence is 0,1,.1.,3,.4.,11,.15.,41,.56.; the m = 3 sequence is 0,1,.1.,4,.5.,19,.24.,91,.115. (dots are to set apart the 2i terms).
2*.1. + 3*.3. + 3*.8. + 1*.21. = 56 the 8th term of m=2 sequence.
2*.1.+ 3*.4.+ 3*.15.+ 1*.56. = 115 the 8th term of the m = 3 sequence
2 + 3*5 + 3*24 + 115 = 204 the 8th term of the m = 4 sequence.
In general the nth row are the coefficients that multiply the 2i terms of the m = j sequence to obtain the 2n th term of the m = j+1 sequence. Since the 2n th term equals the sum of the 2n-1 th and 2n-2 th terms, it follows that the difference between the n th and n-1 rows essencially gives coefficients multiplying 2i terms to add up to the 2n-1 th term. This is so with the following condition: instead of subtracting 1 from the the term preceeding the nth coefficient of the nth row, the 2n-1 term is added instead of the 2n th term. Thus to obtain the 7th term of the m = j+1 sequence the coefficients are respectively 2-1, 3-2, 3 and 1 respectively multiplying the 2nd, 4th, 6th and 7th terms of the m = j sequence. Similarly coefficients for obtaining any 2n-1 term of a j+1 sequence are determined from the nth and n-1 th rows of the above triangle.
So number theory may be important in the study of RNA type molecules.
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