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A144128
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Chebyshev U(n,x) polynomial evaluated at x=18.
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23
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1, 36, 1295, 46584, 1675729, 60279660, 2168392031, 78001833456, 2805897612385, 100934312212404, 3630829342034159, 130608922001017320, 4698290362694589361, 169007844135004199676, 6079584098497456598975
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OFFSET
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1,2
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COMMENTS
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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)
a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,35}. - Milan Janjic, Jan 26 2015
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LINKS
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FORMULA
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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_{k=0..floor(n/2)} (-1)^k*binomial(n-1-k, k)*36^(n-1-2*k). (End)
Product {n >= 1} (1 + 1/a(n)) = 1/17*(17 + sqrt(323)). - Peter Bala, Dec 23 2012
Product {n >= 2} (1 - 1/a(n)) = 1/36*(17 + sqrt(323)). - Peter Bala, Dec 23 2012
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MAPLE
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seq( simplify(ChebyshevU(n, 18)), n=0..20); # G. C. Greubel, Dec 22 2019
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MATHEMATICA
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PROG
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(Sage) [lucas_number1(n, 36, 1) for n in range(1, 16)] # Zerinvary Lajos, Nov 07 2009
(PARI) a(n) = polchebyshev(n, 2, 18); \\ Michel Marcus, Feb 09 2018
(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]]; // Bruno Berselli, Nov 21 2011
(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
(Maxima) makelist(sum((-1)^k*binomial(n-1-k, k)*36^(n-1-2*k), k, 0, floor(n/2)), n, 1, 15); \\ Bruno Berselli, Nov 21 2011
(GAP) a:=[1, 36];; for n in [3..20] do a[n]:=36*a[n-1]-a[n-2]; od; a; # Muniru A Asiru, Feb 09 2018
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CROSSREFS
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Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077412 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), this sequence (m=18), A078987 (m=19), A097316 (m=33).
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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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As Michel Marcus points out, some parts of this entry assume the offset is 1, others parts assume the offset is 0. The whole entry needs careful editing. - N. J. A. Sloane, Feb 10 2018
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STATUS
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approved
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