Forwarded from دستیار زیر نویس و هایپر لینک
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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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.
@infinitymath
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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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
@infinitymath
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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~ Paul Dirac
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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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~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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~W. K. Clifford Common Sense in the Exact Sciences
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