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A144128
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Chebyshev U(n,x) polynomial evaluated at x=18.
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4
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1, 36, 1295, 46584, 1675729, 60279660, 2168392031, 78001833456, 2805897612385, 100934312212404, 3630829342034159, 130608922001017320, 4698290362694589361, 169007844135004199676, 6079584098497456598975
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Contribution from Bruno Berselli, Nov 21 2011: (Start)
A Diophantine property of these numbers: ((a(n+1)-a(n-1))/2)^2 - 323*a(n)^2 = 1.
More generally, for t(m)=m+sqrt(m^2-1) and u(n)=(t(m)^n-1/t(m)^n)/(t(m)-1/t(m)), we can verify that ((u(n+1)-u(n-1))/2)^2-(m^2-1)*u(n)^2=1. (End)
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for sequences related to linear recurrences with constant coefficients, signature (36,-1).
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FORMULA
| Contribution from Bruno Berselli, Nov 21 2011: (Start)
G.f.: x/(1-36*+x^2).
a(n) = 36*a(n-1)-a(n-2) with a(1)=1, a(2)=36.
a(n) = (t^n-1/t^n)/(t-1/t) for t = 18+sqrt(323).
a(n) = sum((-1)^k*binomial(n-1-k, k)*36^(n-1-2k), k=0..floor(n/2)). (End)
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*35^k. - DELEHAM Philippe, Feb 10 2012
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MATHEMATICA
| lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 18]], {n, 0, 8^2}]; lst
LinearRecurrence[{36, -1}, {1, 36}, 20] (* Vincenzo Librandi, Nov 22 2011 *)
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PROG
| (Other) sage: [lucas_number1(n, 36, 1) for n in xrange(1, 16)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 07 2009]
Contribution from Bruno Berselli, Nov 21 2011: (Start)
(PARI) Vec(x/(1-36*x+x^2)+O(x^16))
(MAGMA) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-323); S:=[((18+r)^n-1/(18+r)^n)/(2*r): n in [1..15]]; [Integers()!S[j]: j in [1..#S]];
(Maxima) makelist(sum((-1)^k*binomial(n-1-k, k)*36^(n-1-2*k), k, 0, floor(n/2)), n, 1, 15); (End)
(MAGMA) I:=[1, 36]; [n le 2 select I[n] else 36*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 22 2011
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CROSSREFS
| Cf. A029547, A029548, A200441, A200442, A200724 (incomplete list).
Sequence in context: A170717 A170755 A158700 * A009980 A041613 A203333
Adjacent sequences: A144125 A144126 A144127 * A144129 A144130 A144131
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 11 2008
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