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A262027
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The positive fundamental solutions x = x0(n) for the Pell equation x^2 - d*y^2 = +1 with odd y = y0(n). Then d coincides with d(n) = A007970(n).
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3
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2, 8, 3, 10, 4, 170, 24, 5, 26, 1520, 17, 6, 19, 3482, 48, 7, 50, 530, 8, 48842, 3480, 26, 80, 9, 82, 28, 197, 1574, 49, 10, 227528, 51, 962, 1126, 120, 11, 122, 4730624, 577, 10610, 244, 35, 77563250, 12, 1728148040, 37, 1324, 721, 64080026, 168, 13, 170, 2024, 199, 4190210
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OFFSET
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1,1
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COMMENTS
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The corresponding values y = y0(n) are given by A262026(n).
This is a proper subset of A033313 corresponding to D values from d(n) = A007970(n).
For the proof that d(n) = A007970(n), the products of Conway's 2-happy couples, see the W. Lang link under A007970.
If d(n) = A007970(n) is odd (necessarily congruent to 3 modulus 4) then x0(n) is even, and if d(n) is even (necessarily congruent to 0 modulus 8) then x0 is odd.
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LINKS
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FORMULA
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a(n)^2 - d(n)*y0(n)^2 = +1 with y0(n) = A262026(n) and d(n) = A007970(n). (x0(n) = a(n), y0(n)) are the positive fundamental solutions of this Pell equation x^2 - d*y^2 = +1 with odd y = y0.
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EXAMPLE
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For the first [d(n), x0(n), y0(n)] see A262026.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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