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A296376
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Natural numbers x such that 7*y^2 = x^2 + x + 1 has a solution in natural numbers.
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3
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2, 18, 653, 4701, 165986, 1194162, 42159917, 303312573, 10708453058, 77040199506, 2719904916941, 19567907362077, 690845140450082, 4970171429768178, 175471945769404013, 1262403975253755261, 44569183380288169346, 320645639543024068242, 11320397106647425609997, 81442730039952859578333
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OFFSET
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1,1
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COMMENTS
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Given explicitly as the numerators of the convergents to the continued fractions
[2,(1,1,1,4)^i,5,(1,1,1,4)^{i-1},1,2] (for n odd and i = (n-1)/2)
and
[2,(1,1,1,4)^i,1,1,2,(1,4,1,1)^i,1] (for n even and i = n/2 - 1).
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REFERENCES
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E.-A. Majol, Note #2228, L'Intermédiaire des Mathématiciens, 9 (1902), pp. 183-185. - N. J. A. Sloane, Mar 02 2022
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LINKS
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FORMULA
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a(n) = 255*a(n-2) - 255*a(n-4) + a(n-6).
G.f.: x*(2 + 16*x + 127*x^2 - 16*x^3 - 3*x^4) / ((1 - x)*(1 - 16*x + x^2)*(1 + 16*x + x^2)).
a(n) = a(n-1) + 254*a(n-2) - 254*a(n-3) - a(n-4) + a(n-5) for n>5.
(End)
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EXAMPLE
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For n = 3 the pair is (x,y) = (653,247).
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MAPLE
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f:= gfun:-rectoproc({a(n) = 255*a(n-2) - 255*a(n-4) + a(n-6), a(1)=2, a(2)=18, a(3)=653, a(4)=4701, a(5)= 165986, a(6)=1194162}, a(n), remember):
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PROG
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(PARI) Vec(x*(2 + 16*x + 127*x^2 - 16*x^3 - 3*x^4) / ((1 - x)*(1 - 16*x + x^2)*(1 + 16*x + x^2)) + O(x^25)) \\ Colin Barker, Dec 13 2017
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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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EXTENSIONS
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STATUS
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approved
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