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I am asking for books/any other online sources with the following philosphy$^{[1]}$:

This is a mathematics, not a programming book. It's intended for students, mathematics teachers and mathematicians who are just starting to explore mathematics on their own computers. In studying it, and especially working through the problems, readers will get to know many new elementary topics and learn as much from the extensive exercises as from the examples

Which basically means you have some problems, and with your computers you find and discover new patterns, which are tedious to do by hand, and then you prove them without computer.

Notes :
$^{[1]}$ The only book I know is Arthur Engel's "Exploring mathematics with your computer" , but it's not available at a reasonable price. The text is copied from it's preface, from the preview at Amazon. I forgot to mention, but I know Project Euler too, and it moderately fits to my criterion.

$[2]$ Related to my this question .

$[3]$ Books like "A=B" doesn't quite fits to my criterion, because you don't prove stuff with hypergeometric series there, the computer proves it for you. But "Computational recreations in mathematica" (by Illan Vardi) fits to my criterion, because you prove the stuff yourself.

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    $\begingroup$ While not being a book, Project Euler is good for number theoretical stuff. $\endgroup$
    – Bobson Dugnutt
    Commented Jun 6, 2017 at 8:29
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    $\begingroup$ @lovsovs PE doesn't really teach until you've already solved the problem and look at the posts $\endgroup$
    – qwr
    Commented Jun 6, 2017 at 8:36
  • $\begingroup$ "A = B" deals with hypergeometric identities and computer use $\endgroup$
    – qwr
    Commented Jun 6, 2017 at 9:08
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    $\begingroup$ Some of these might be of interest: experimentalmath.info/books $\endgroup$
    – Hans Lundmark
    Commented Jun 6, 2017 at 10:54
  • $\begingroup$ @qwr I'd still classify that as educational. $\endgroup$
    – Bobson Dugnutt
    Commented Jun 6, 2017 at 11:04

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The following quote is the starting point of chapter I in

  • Mathematics by Experiment, Plausible Reasoning in the 21st Century by J. Borwein and D. Bailey

    ''The computer has in turn changed the very nature of mathematical experience, suggesting for the first time that mathematics, like physics, may yet become an empirical discipline, a place where things are discovered because they are seen.''

    From: David Berlinski, ''Ground Zero: A Review of the The Pleasures of Counting, by T.W. Koerner,'' 1997

Both authors are well known for their highly creative usage of computational power in mathematical problems.

  • From the preface of the book:

    Our goal in these books is to pesent a variety of accessible examples of modern mathematics where intelligent computing plays a significant role (along with a few examples showing the limitations of computing). We have concentrated primarily on examples from analysis and number theory, as this is where we have the most experience but there are numerous excursions into other areas of mathematics as well.

Here the authors also address the follow-up book: Experimentation in Mathematics - Computational Paths to Discovery.

Hint: You might also want to take a look at experimentalmath.info.

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  • $\begingroup$ Hint: Did you check the comment section of the question, especially the comment by Hans ? :P $\endgroup$
    – user403711
    Commented Jun 16, 2017 at 7:10
  • $\begingroup$ @ArbitraryKangaroo: No, but I've checked the link in your note section. Considering the stated preface in my answer I think these books might fit your needs. $\endgroup$
    – Markus Scheuer
    Commented Jun 16, 2017 at 8:03
  • $\begingroup$ Yeah, sure, definitely it fits, thanks, but its a bit high level for me. :( $\endgroup$
    – user403711
    Commented Jun 16, 2017 at 8:27
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Ilan Vardi, Computational Recreations in Mathematica.

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A = B by Marko Petkovsek, Herbert S Wilf, & Doron Zeilberger

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  • $\begingroup$ Check the updated question. $\endgroup$
    – user403711
    Commented Jun 15, 2017 at 2:57
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You might like Coding the Matrix, Linear Algebra through Applications to Computer Science. There is also a MOOC for the course on Coursera.

The following excerpt is from the website:

The aim of this course is to provide students interested in computer science an introduction to vectors and matrices and their use in CS applications.

The course is driven by applications from areas chosen from among: computer vision, cryptography, game theory, graphics, information retrieval and web search, and machine learning.

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http://www.math.tamu.edu/~harold.boas/courses/math696/Maple.html

See the lecture on "slicing bread". It uses Maple to explore some concepts in higher-dimensional Euclidean geometry. (The other lectures provide the necessary Maple background).

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