The Ulam sequence is a sequence of positive integers a_n, where a_1=1, a_2=2, and where each a_n for n > 2 is defined to be the smallest integer that can be expressed as the sum of two distinct earlier terms in a unique way.

The first few terms of the sequence are shown in the picture: 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47. The third term is 3, because 3=1+2. The fourth term is 4, because although 4 can be expressed in two ways as the sum of two earlier terms (1+3, or 2+2) the sum “2+2” is not a sum of distinct earlier terms. The fifth term is not 5, because 5=1+4=2+3; rather, it is 6, which is 2+4. The sixth term is not 7, because 7=1+6=3+4, but rather 8, which is 2+6. And so on.

The Ulam sequence is named after the Polish-American mathematician Stanisław Ulam (1909–1984) who introduced it in 1964 in a survey on unsolved problems. Ulam remarked that it can be notoriously difficult to answer questions about the properties of sequences like this one, even if they are defined by a simple rule.

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Geometry is the science of 
correct reasoning on incorrect figures.

~George Polya
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There are certainly people who regard √2 as something perfectly obvious but jib at √-1.
This is because they think they can visualise  the former as something in physical space  but not the latter. 
Actually √-1 is a much simpler concept. 

~Edward Titschmarsh
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 I learned to distrust all physical concepts as the basis for a theory. Instead one should put one's trust in a mathematical scheme, even if the scheme does not appear at first sight to be connected with physics. One should concentrate on getting interesting mathematics. 

~ Paul Dirac
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برای اطلاعات بیشتر به سایت انجمن ریاضی ایران مراجعه کنید.
http://fa.ims.ir

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موسیقی ، ریاضی ذهن است
و ریاضی ، موسیقی ذهن.....
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نموداری از روشِ پیشنهادی از سوی ابوريحان بیرونی برای برآوردِ شعاع و دورادورِ زمین
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No mathematician can be a complete mathematician unless he is also something of a poet.

~Karl Weierstrass
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We may always depend on it that algebra, which cannot be translated into good English and sound common sense, is bad algebra.

~W. K. Clifford Common Sense in the Exact Sciences
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The friendship paradox is the observation that your friends, on average, have more friends than you do. This phenomenon, which was first observed by the sociologist Scott L. Feld in 1991, is mathematically provable, even though it contradicts most people's intuition that they have more friends than their friends do.

The recent paper The Majority Illusion in Social Networks (http://arxiv.org/abs/1506.03022) by Kristina Lerman, Xiaoran Yan, and Xin-Zeng Wu explores some phenomena that are related to the friendship paradox. The authors explain how, under certain conditions, the structure of a social network can make it appear to an individual that certain types of behaviour are far more common than they actually are.
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If you open a mathematics paper at random, on the pair of pages before you, you will find a mistake.

~Joseph Doob
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