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Search: id:A141326
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| A141326 |
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Subsequence of 'Fermat near misses' which is generated by a simple formula based on the cubic binomial expansion along with formulae for the corresponding terms in the expression, x^3 + y^3 = z^3 + 1. |
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4
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| 12, 150, 738, 2316, 5640, 11682, 21630, 36888, 59076, 90030, 131802, 186660, 257088, 345786, 455670, 589872, 751740, 944838, 1172946, 1440060
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Note that the given formulae generate 1 + 12^3 = 9^3 + 10^3 = 1729 for n=1. In this case b(1) < c(1), whereas b(n) > c(n) for n > 1.
Contribution from Lewis Mammel (l_mammel(AT)att.net), Aug 21 2008: (Start)
In Ramanujan's parametric equation:
(ax+y)^3 + (b+x^2y)^3 = (bx+y)^3 + (a+x^2y)^3
where
a^2 + ab + b^2 = 3xy^2
This sequence is obtained by setting a=0, y=1 and finding the solution to b^2=3x:
b=3n, x=3n^2 (End)
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LINKS
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Wolfram Mathworld, Diophantine Equation--3rd Powers [From Lewis Mammel (l_mammel(AT)att.net), Aug 21 2008]
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FORMULA
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With a(n) = 9*n^4 + 3*n b(n) = 9*n^4 c(n) = 9*n^3 + 1 1 + a(n)^3 = b(n)^3 + c(n)^3, by substitution and expansion
With a(n) = 9*n^4 + 3*n, b(n) = 9*n^4 and c(n) = 9*n^3 + 1, we have 1 + a(n)^3 = b(n)^3 + c(n)^3, by substitution and expansion [From Lew Mammel (l_mammel(AT)att.net), Aug 09 2008]
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CROSSREFS
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Cf. A050791, A050792, A050793, A050794.
Sequence in context: A057572 A114106 A015610 this_sequence A154733 A056351 A056345
Adjacent sequences: A141323 A141324 A141325 this_sequence A141327 A141328 A141329
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KEYWORD
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easy,nonn
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AUTHOR
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Lewis Mammel (l_mammel(AT)att.net), Aug 03 2008
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