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I found a project that is very clever and I would like to build my own but using some of their ideas. That's basically what I did but now the code is similar, because we follow the same idea, but I did not reuse any of their code, I've handwritten it entirely, but now it looks similar.
Do I have to mention where the original idea come from, and include their copyright license, even if I did not use any of their code? I have just written mine myself but it looks similar.
The right to produce adaptations of a copyrighted work (aka derivative works) is one of the rights reserved to the copyright holder in Berne Convention countries. The rightsholder may usually licence it to others if (s)he chooses to do so.
You've read this program, then set out to produce your own program that implements the same ideas, and by your own admission "it looks similar" to the original work. I can't say with certainty whether or not what you've produced is an adaptation of the original work, but I wouldn't like to bet against it. So let's assume that it is, for the rest of this answer.
Fortunately, the original work is licensed under 3BSD, which says that "redistribution and use in source and binary forms, with or without modification, are permitted". Since your work is an adaptation of the original, you are distributing the original work, "with modification". Three conditions are given for this:
Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.
Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.
Neither the name of the copyright holder nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.
It doesn't sound like you were planning to contravene (3), so that issue doesn't arise. You haven't told us whether you'll be distributing your project in source or binary form, but when you do, the conditions that you must satisfy are clearly spelled out above.
