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A097843

First differences of Chebyshev polynomials S(n,123) = A049670(n+1) with Diophantine property.
(history; published version)

#48 by Joerg Arndt at Sun Dec 31 11:28:12 EST 2023

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editing

approved

#47 by Paolo P. Lava at Sun Dec 31 10:35:36 EST 2023

FORMULA

a(n) = (1/2)*((123/2 - (55/2)*sqrt(5))^n + (123/2 + (55/2)*sqrt(5))^n) + (11/50)*sqrt(5)*((123/2 + (55/2)*sqrt(5))^n - (123/2 - (55/2)*sqrt(5))^n), with n >= 0. - Paolo P. Lava, Dec 12 2008

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editing

#46 by Charles R Greathouse IV at Thu Sep 08 08:45:14 EDT 2022

PROG

(MAGMAMagma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)/(1-123*x+x^2) )); // G. C. Greubel, Jan 14 2019

Discussion

Thu Sep 08

08:45

OEIS Server: https://oeis.org/edit/global/2944

#45 by Giovanni Resta at Thu Jan 23 03:45:25 EST 2020

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#44 by Michel Marcus at Thu Jan 23 02:09:28 EST 2020

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proposed

#43 by Michel Marcus at Thu Jan 23 02:09:25 EST 2020

LINKS

H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html ">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume.

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#42 by Jon E. Schoenfield at Thu Jan 23 00:06:17 EST 2020

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proposed

#41 by Jon E. Schoenfield at Thu Jan 23 00:06:02 EST 2020

FORMULA

a(n) = ((-1)^n)*S(2*n, 11*Ii) with the imaginary unit I i and the S(n, x) = U(n, x/2) Chebyshev polynomials.

a(n) = 123*a(n-1) - a(n-2) for n > 1, a(0)=1, a(1)=122. - Philippe Deléham, Nov 18 2008

a(n) = (1/2)*{[((123/2) - (55/2)*sqrt(5)])^n + [(123/2) + (55/2)*sqrt(5)])^n} ) + (11/50)*sqrt(5)*{[((123/2) + (55/2 )*sqrt(5)])^n - [(123/2) - (55/2)*sqrt(5)])^n}, ), with n >= 0. - Paolo P. Lava, Dec 12 2008

a(n) = (1/5)*F(10*n + 5). sum Sum_{n >= 1} 1/( a(n) - 1/a(n) ) = 1/11^2. Compare with A001519 and A007805. - Peter Bala, Nov 29 2013

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approved

editing

#40 by Susanna Cuyler at Fri Apr 19 13:41:03 EDT 2019

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reviewed

approved

#39 by Michel Marcus at Fri Apr 19 13:31:15 EDT 2019

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proposed

reviewed