What are some algorithms for finding a closed form function given an integer sequence?

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I\'m looking form a programatic way to take an integer sequence and spit out a closed form function. Something like:

Given: 1,3,6,10,15

Return: n(n+1)/2

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伪装坚强ぢ

2020-12-30 15:31

There is no one function in general.

For the sequence you specified, The On-Line Encyclopedia of Integer Sequences finds 133 matches in its database of interesting integer sequences. I've copied the first 5 here.

A000217 Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431

A130484 Sum {0<=k<=n, k mod 6} (Partial sums of A010875).
0, 1, 3, 6, 10, 15, 15, 16, 18, 21, 25, 30, 30, 31, 33, 36, 40, 45, 45, 46, 48, 51, 55, 60, 60, 61, 63, 66, 70, 75, 75, 76, 78, 81, 85, 90, 90, 91, 93, 96, 100, 105, 105, 106, 108, 111, 115, 120, 120, 121, 123, 126, 130, 135, 135, 136, 138, 141, 145, 150, 150, 151, 153

A130485 Sum {0<=k<=n, k mod 7} (Partial sums of A010876).
0, 1, 3, 6, 10, 15, 21, 21, 22, 24, 27, 31, 36, 42, 42, 43, 45, 48, 52, 57, 63, 63, 64, 66, 69, 73, 78, 84, 84, 85, 87, 90, 94, 99, 105, 105, 106, 108, 111, 115, 120, 126, 126, 127, 129, 132, 136, 141, 147, 147, 148, 150, 153, 157, 162, 168, 168, 169, 171, 174, 178, 183

A104619 Write the natural numbers in base 16 in a triangle with k digits in the k-th row, as shown below. Sequence gives the leading diagonal.
1, 3, 6, 10, 15, 2, 1, 1, 14, 3, 2, 2, 5, 12, 4, 4, 4, 13, 6, 7, 11, 6, 9, 9, 10, 7, 12, 13, 1, 0, 1, 10, 5, 1, 12, 8, 1, 1, 14, 1, 9, 7, 1, 4, 3, 1, 2, 2, 1, 3, 4, 2, 7, 9, 2, 14, 1, 2, 8, 12, 2, 5, 10, 3, 5, 11, 3, 8, 15, 3, 14, 6, 3, 7, 0, 4, 3, 13, 4, 2, 13, 4, 4, 0, 5, 9, 6, 5, 1, 15, 5, 12, 11, 6

A037123 a(n) = a(n-1) + Sum of digits of n.
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 46, 48, 51, 55, 60, 66, 73, 81, 90, 100, 102, 105, 109, 114, 120, 127, 135, 144, 154, 165, 168, 172, 177, 183, 190, 198, 207, 217, 228, 240, 244, 249, 255, 262, 270, 279, 289, 300, 312, 325, 330, 336, 343, 351, 360, 370, 381

If you restrict yourself to polynomial functions, this is easy to code up, and only mildly tedious to solve by hand.

Let $f(x)=a_0+a_1x+a_2x^2+\cdots+a_{n-1}x^{n-1}+a_nx^n$ , for some unknown $a_0\ldots a_n$

Now solve the equations
$y_0=f(0)=a_0$
$y_1=f(1)=a_0+a_1+a_2+\cdots+a_{n-1}+a_n$
$y_2=f(2)=a_0+2a_1+4a_2+\cdots+2^{n-1}a_{n-1}+2^na_n$
…
$y_n=f(n)=a_0+na_1+n^2a_2+\cdots+n^{n-1}a_{n-1}+n^na_n$
which simply a system of linear equations.

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