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A097313
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Chebyshev polynomials of the second kind, U(n,x), evaluated at x=15.
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21
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0, 1, 30, 899, 26940, 807301, 24192090, 724955399, 21724469880, 651009141001, 19508549760150, 584605483663499, 17518655960144820, 524975073320681101, 15731733543660288210, 471427031236487965199, 14127079203550978667760
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OFFSET
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-1,3
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COMMENTS
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b(n+1)^2 - 14*(4*a(n))^2 = +1, n>=-1, with b(n)=A068203(n) gives all nonnegative integer solutions of this D=224 Pell equation.
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 30's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
About first comment, more generally, for t(m) = m + sqrt(m^2-1) and u(n) = (t(m)^(n+1) - 1/t(m)^(n+1))/(t(m) - 1/t(m)), we can verify that ((u(n+1) - u(n-1))/2)^2 - (m^2-1)*u(n)^2 = 1. In this case is m=15. - Bruno Berselli, Nov 21 2011
For n>=0, a(n) equals the number of 01-avoiding words of length n on alphabet {0,1,...,29}. - Milan Janjic, Jan 26 2015
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LINKS
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FORMULA
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a(n) = S(n, 30) = U(n, 15), n>=-1, with Chebyshev polynomials of 2nd kind. See A049310 for the triangle of S(n, x) coefficients. S(-1, x) := 0 =: U(-1, x).
G.f.: 1/(1-30*x+x^2).
a(n) = ((15+4*sqrt(14))^(n+1) - (15-4*sqrt(14))^(n+1))/(8*sqrt(14)) (Binet form).
a(n) = sqrt((A068203(n+1)^2 - 1)/224), n>=-1.
a(n) = 30*a(n-1)-a(n-2) for n>0; a(-1)=0, a(0)=1. - Philippe Deléham, Nov 18 2008
With an offset of 0, product {n >= 1} (1 + 1/a(n)) = 1/7*(7 + 2*sqrt(14)). - Peter Bala, Dec 23 2012
Product {n >= 2} (1 - 1/a(n)) = 1/15*(7 + 2*sqrt(14)). - Peter Bala, Dec 23 2012
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MAPLE
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0, seq(orthopoly[U](n, 15), n=0..50); # Robert Israel, Jan 26 2015
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MATHEMATICA
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PROG
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(Sage) [lucas_number1(n, 30, 1) for n in range(0, 20)] # Zerinvary Lajos, Jun 27 2008
(Sage) [chebyshev_U(n, 15) for n in (-1..20)] # G. C. Greubel, Dec 22 2019
(Magma) I:=[0, 1]; [n le 2 select I[n] else 30*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 12 2012
(PARI) vector( 21, n, polchebyshev(n-2, 2, 15) ) \\ G. C. Greubel, Dec 22 2019
(GAP) m:=15;; a:=[0, 1];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 22 2019
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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), this sequence (m=15), A029548 (m=16), A029547 (m=17), A144128 (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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STATUS
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approved
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