Georgi Guninski on Fri, 20 Jan 2023 08:21:42 +0100

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Computing integer sequences via integrals

I am trying to compute integer sequences via closed
forms of definite integrals, but probably this is impossible.

The question is: which integer sequences can be computed
by integrals?

Partial results:

Trying to "discretize" the reals, we are using floor()
and the area of the definite integrals.

For a start consider the plot of floor(x).
? ploth(x=1,10,floor(x))

The area is the union of rectangles and for integer
bounds the area is integer.

sagemath computes the following indefinite integrals:
sage: integrate(floor(x),x)
1/2*(2*x - floor(x) - 1)*floor(x)
sage: integrate(floor(x)^2,x)

The area for floor(x) and the sum(i=1,N-1,x) appear to be computed correctly:
sage: I1=integrate(floor(x),x)
sage: N=13;[I1(x=N)-I1(x=1)-N*(N-1)/2]

The area for floor(x)^2 doesn't work for me, why?

floor() might not be elementary function, but there is closed
form for it using exp() and log(), taking the principal
branch of the logarithm:

? floor1(x)=x - 1/2*I*log(-exp(-2*I*Pi*x))/Pi - 1/2
%12 = (x)->x-1/2*I*log(-exp(-2*I*Pi*x))/Pi-1/2
? floor1(13.1)
%13 = 13.000000000000000000000000000000000000 + 0.E-39*I