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 A097842 Chebyshev polynomials S(n,123) + S(n-1,123) with Diophantine property. 5
 1, 124, 15251, 1875749, 230701876, 28374454999, 3489827263001, 429220378894124, 52790616776714251, 6492816643156958749, 798563656491529211876, 98216836931814936101999, 12079872378956745611334001, 1485726085774747895257980124, 182732228677915034371120221251, 22474578401297774479752529233749 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS (11*a(n))^2 - 5*(5*b(n))^2 = -4 with b(n)=A097843(n) give all positive solutions of this Pell equation. LINKS Indranil Ghosh, Table of n, a(n) for n = 0..477 Tanya Khovanova, Recursive Sequences H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277. H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume Index entries for linear recurrences with constant coefficients, signature (123,-1). FORMULA a(n) = S(n, 123) + S(n-1, 123) = S(2*n, 5*sqrt(5)), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x). S(n, 123)=A049670(n+1). a(n) = (-2/11)*I*((-1)^n)*T(2*n+1, 11*I/2) with the imaginary unit I and Chebyshev's polynomials of the first kind. See the T-triangle A053120. G.f.: (1+x)/(1-123*x+x^2). a(n) = 123*a(n-1)-a(n-2) for n>1 ; a(0)=1, a(1)=124 . [Philippe Deléham, Nov 18 2008] From Peter Bala, Mar 23 2015: (Start) a(n) = ( Fibonacci(10*n + 10 - 2*k) + Fibonacci(10*n + 2*k) )/( Fibonacci(10 - 2*k) + Fibonacci(2*k) ), for k an arbitrary integer. a(n) = ( Fibonacci(10*n + 10 - 2*k - 1) - Fibonacci(10*n + 2*k + 1) )/( Fibonacci(10 - 2*k - 1) - Fibonacci(2*k + 1) ), for k an arbitrary integer, k != 2. The aerated sequence (b(n))n>=1 = [1, 0, 124, 0, 15251, 0, 1875749, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -121, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials. (End) a(n) = Lucas(10*n + 5)/11. - Ehren Metcalfe, Jul 29 2017 EXAMPLE All positive solutions of Pell equation x^2 - 125*y^2 = -4 are (11=11*1,1), (1364=11*124,122), (167761=11*15251,15005), (20633239=11*1875749,1845493), ... MATHEMATICA CoefficientList[Series[(1 + x)/(1 - 123 x + x^2), {x, 0, 11}], x] (* Michael De Vlieger, Feb 08 2017 *) PROG (PARI) a(n)=polchebyshev(n, 2, 123/2) + polchebyshev(n - 1, 2, 123/2); \\ Michel Marcus, Aug 04 2017 CROSSREFS Sequence in context: A289299 A035816 A206077 * A206189 A280905 A146516 Adjacent sequences:  A097839 A097840 A097841 * A097843 A097844 A097845 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Sep 10 2004 STATUS approved

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