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A077422
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Chebyshev sequence T(n,11) with Diophantine property.
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10
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1, 11, 241, 5291, 116161, 2550251, 55989361, 1229215691, 26986755841, 592479412811, 13007560326001, 285573847759211, 6269617090376641, 137646002140526891, 3021942430001214961, 66345087457886202251, 1456569981643495234561, 31978194508699008958091
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OFFSET
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0,2
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COMMENTS
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Except for the first term, positive values of x (or y) satisfying x^2 - 22xy + y^2 + 120 = 0. - Colin Barker, Feb 19 2014
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LINKS
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FORMULA
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a(n+1)^2 - 30*(2*b(n))^2 = 1, n>=0, with the companion sequence b(n)=A077421(n).
a(n) = 22*a(n-1) - a(n-2), a(-1) := 11, a(0)=1.
a(n) = T(n, 11) = (S(n, 22)-S(n-2, 22))/2 = S(n, 22)-11*S(n-1, 22) with T(n, x), resp. S(n, x), Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(n, 22)=A077421(n).
a(n) = (ap^n + am^n)/2 with ap := 11+2*sqrt(30) and am := 11-2*sqrt(30).
a(n) = sum(((-1)^k)*(n/(2*(n-k)))*binomial(n-k, k)*(2*11)^(n-2*k), k=0..floor(n/2)), n>=1.
a(n+1) = sqrt(1 + 30*(2*A077421(n))^2), n>=0.
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MATHEMATICA
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LinearRecurrence[{22, -1}, {1, 11}, 20] (* Harvey P. Dale, Jul 30 2022 *)
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PROG
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(Sage) [lucas_number2(n, 22, 1)/2 for n in range(0, 20)] # Zerinvary Lajos, Jun 26 2008
(Magma) [n: n in [1..10000000] |IsSquare(30*(n^2-1))] // Vincenzo Librandi, Aug 08 2010
(PARI) Vec((1-11*x)/(1-22*x+x^2) + O(x^100)) \\ Colin Barker, Jun 15 2015
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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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STATUS
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approved
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