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A089270
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Positive numbers represented by the integer binary quadratic form x^2 + x*y - y^2 with relative prime x and y.
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11
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1, 5, 11, 19, 29, 31, 41, 55, 59, 61, 71, 79, 89, 95, 101, 109, 121, 131, 139, 145, 149, 151, 155, 179, 181, 191, 199, 205, 209, 211, 229, 239, 241, 251, 269, 271, 281, 295, 305, 311, 319, 331, 341, 349, 355, 359, 361, 379, 389, 395, 401, 409, 419, 421, 431
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| The negative numbers represented by x^2 + x*y - y^2 with relative prime x and y are -a(n).
The discriminant of this binary form is D=5>0, hence this is an indefinite form.
It appears that these are also the n for which the equation x^2 = x+1 (mod n) has solutions. The number of solutions is 0 or a power of 2. It appears that n=5 is the only n for which x^2 = x+1 (mod n) has just one solution. The first n producing 4 solutions is 209. The first n producing 8 solutions is 6061. [From T. D. Noe (noe(AT)sspectra.com), Nov 04 2009]
The terms are the product of primes congruent to {0,1,4} mod 5, which is A038872.
Brother Alfred's paper lists these numbers (less than 1000) as discriminants of Fibonacci sequences. For each number, he also lists the (a,b) pairs that are the first two terms of a unique Fibonacci sequence.
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REFERENCES
| Brother Alfred Brousseau, On the ordering of Fibonacci sequences, Fib. Quart. 1.4 (1963), 43-46; Errata, Fib. Quart. 2.1 (1964), 38.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
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FORMULA
| a(n) = x^2 + x*y - y^2 with relative prime integers x and y (proper solutions of the Diophantine equation).
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EXAMPLE
| n=2: a(2)=5 with, for example, (x,y)= (2,1): 4+2-1=5 (there are infinitely many proper (x,y) solutions).
n=8: a(n)=55 with, for example, (x,y)=(7,6) or (7,1). In this case there exist two fundamental proper solutions.
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CROSSREFS
| Sequence in context: A051349 A048217 A132087 * A038872 A141158 A130828
Adjacent sequences: A089267 A089268 A089269 * A089271 A089272 A089273
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KEYWORD
| nonn
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Nov 07 2003
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