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Search: id:A077446
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| A077446 |
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2*n^2 + 14 is a square. |
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+0
2
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| 1, 5, 11, 31, 65, 181, 379, 1055, 2209, 6149, 12875, 35839, 75041, 208885, 437371, 1217471, 2549185, 7095941, 14857739, 41358175, 86597249, 241053109, 504725755, 1404960479, 2941757281, 8188709765, 17145817931, 47727298111
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OFFSET
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1,2
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COMMENT
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The equation "2*n^2 + 14 is a square" is a version of the generalized Pell Equation x^2 - D*y^2 = C where x^2 - 2*y^2 = 14.
Numbers n such that (ceiling(sqrt(n*n/2)))^2 = (7+n*n)/2 [From Ctibor O. Zizka (c.zizka(AT)email.cz), Nov 09 2009]
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REFERENCES
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A. H. Beiler, "The Pellian." Ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.
L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, p. 341-400.
Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, p. 139-147.
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LINKS
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J. J. O'Connor and E. F. Robertson, Pell's Equation
Eric Weisstein's World of Mathematics, ; Pell Equation
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FORMULA
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Lim. n-> Inf. a(n)/a(n-2) = 5.82842712474619009760 = 3 + 2*SQRT(2) = RG (Great Ratio). Lim. k-> Inf. a(2*k+1)/a(2*k) = 2.09383632135605431360 = (9 + 4*SQRT(2))/7 = R1 (Ratio 1). Lim. k -> Inf. a(2*k)/a(2*k-1) = 2.78361162489122432754 = (11 + 6*SQRT(2))/7 = R2 (Ratio 2); RG = R1*R2. a(n) = 6*a(n-2) - a(n-4).
a(2*k-1) = [ 2*[(3+2*Sqrt(2))^n - (3-2*Sqrt(2))^n] - [(3+2*Sqrt(2))^(n-1) - (3-2*Sqrt(2))^(n-1)] + [(3+2*Sqrt(2))^(n-2) - (3-2*Sqrt(2))^(n-2)] ] / (4*Sqrt(2)) a(2*k) = [ 5*[(3+2*Sqrt(2))^n - (3-2*Sqrt(2))^n] + [(3+2*Sqrt(2))^(n-1) - (3-2*Sqrt(2))^(n-1)] ] / (4*Sqrt(2)). a(n) = 6*a(n-2) - a(n-4)
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CROSSREFS
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2*(a(n))^2 + 14 = (A077447)^2.
Sequence in context: A057470 A038580 A106088 this_sequence A023276 A074648 A106908
Adjacent sequences: A077443 A077444 A077445 this_sequence A077447 A077448 A077449
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KEYWORD
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nonn,new
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AUTHOR
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Gregory V. Richardson (omomom(AT)hotmail.com), Nov 09 2002
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