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What process does Mathematica use to find closed forms for sequences?

Posted 10 years ago

I have a sequence such as

{6 Sqrt[3/17], 54 Sqrt[3/1757], 12 Sqrt[3/113], 42 Sqrt[3/1853], 18 Sqrt[3/473], 6 Sqrt[3/77], 3 Sqrt[6/61], 18 Sqrt[3/1973], 6 Sqrt[3/497], 6 Sqrt[3/1997]},

which I can square to obtain

{108/17, 8748/1757, 432/113, 5292/1853, 972/473, 108/77, 54/61, 972/1973, 108/497, 108/1997},

from which I wish to interpolate to some polynomial/rational form. The "interpolating polynomial" function leaves a ninth order polynomial, but WolframAlpha spits out a rational function with quadratic numerator and denominator:

Subscript[a, n]==-((108 (n^2-22 n+121))/(3 n^2-66 n-1637)),

which I can square root to obtain an approximate closed form for the original sequence. How does it do this?

Furthermore, what is to be done when the numbers are apparently too large for WolframAlpha's input:

{13/(1156 Sqrt[51]), (2183619 Sqrt[3/1757])/12348196, 2219/(12769 Sqrt[339]), 9778951/(13734436 Sqrt[5559]), (96733 Sqrt[3/473])/894916, 2593/(23716 Sqrt[231]), 3209/(29768 Sqrt[366]), (1637233 Sqrt[3/1973])/15570916, 101293/(988036 Sqrt[1491]), 1605793/(15952036 Sqrt[5991])}?

Thank you

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POSTED BY: Lone Ranger
2 Replies

That's a very useful function; thank you Marco. However, I am still unsure what I may conclude when this does not give a result. e.g for the sequence 19683/78608, 2418947995203/8678312148800, 172687707/577158800, \ 3191302184643/10179963963200, 13677256323/42329526800, \ 9623043/29218112, 60507243/181584800, 4122657892323/12288566907200, \ 16535178243/49105389200, 4298362362243/12742486356800

Are the numbers too large somehow?!

Cheers

POSTED BY: Lone Ranger

Hi,

if you type the squared number sequence in Woflram alpha gives:

enter image description here

In Mathematica the following command does the trick:

FindSequenceFunction[{108/17, 8748/1757, 432/113, 5292/1853, 972/473, 
  108/77, 54/61, 972/1973, 108/497, 108/1997}]

Best wishes, Marco

POSTED BY: Marco Thiel
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