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A080663
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a(n) = 3*n^2 - 1.
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16
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2, 11, 26, 47, 74, 107, 146, 191, 242, 299, 362, 431, 506, 587, 674, 767, 866, 971, 1082, 1199, 1322, 1451, 1586, 1727, 1874, 2027, 2186, 2351, 2522, 2699, 2882, 3071, 3266, 3467, 3674, 3887, 4106, 4331, 4562, 4799, 5042, 5291, 5546, 5807, 6074, 6347, 6626
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OFFSET
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1,1
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COMMENTS
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These numbers cannot be perfect squares. See the Hilliard link for a proof.
2nd elementary symmetric polynomial of n, n + 1 and n + 2: n(n+1) + n(n+2) + (n+1)(n+2). - Zak Seidov, Mar 23 2005
The Diophantine equation x(x+1) + (x+2)(x+3) = (x+y)^2 + (x-y)^2 has solutions x = a(n), y = 3n. - Bruno Berselli, Mar 29 2013
A simpler proof that these numbers can't be perfect squares can easily be constructed using congruences: If the equation x^2 = 3y^2 - 1 has a solution in positive integers, then x^2 = 2 mod 3. Obviously we can't have x = 0 mod 3, and x = 1 mod 3 doesn't work either because then x^2 = 1 mod 3 also. That leaves x = 2 mod 3, but then x^2 = 1 mod 3. - Alonso del Arte, Oct 19 2013
2*a(n+1) is surface area of a rectangular prism with consecutive integer sides: n, n+1, and n+2, (n>0). - Wesley Ivan Hurt, Sep 06 2014
Numbers m such that 3*m+3 is a square. So, these are the numbers m such that the system of equations x=sqrt(m-2yz), y=sqrt(m+1-2xz), z=sqrt(m+2-2xy) admits 3 real positive solutions whose sum is an integer. See the Rechtman link. - Michel Marcus, Jun 06 2020
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REFERENCES
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Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach. Mineola, New York: Dover Publications (1969, reprinted 2007): p. 7, Problem 6.6.
E. Grosswald, Topics from the Theory of Numbers, 1966 p 64 problem 11
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LINKS
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FORMULA
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G.f.: x*(2 + 5*x - x^2)/(1-x)^3. - Joerg Arndt, Sep 06 2014
Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(3)*cot(Pi/sqrt(3)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(3))*csc(Pi/sqrt(3)) - 1)/2.
Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(3))*csc(Pi/sqrt(3)).
Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(3))*sin(sqrt(2/3)*Pi)/sqrt(2). (End)
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MAPLE
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MATHEMATICA
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PROG
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(PARI) list(n) = { for(x=1, n, y = 3*x*x-1; print1(y, ", ") ) } \\ edited by Michel Marcus, Feb 01 2020
(PARI) Vec(x*(2+5*x-x^2)/(1-x)^3+O(x^66)) \\ Joerg Arndt, Sep 06 2014
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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