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A049670 a(n) = Fibonacci(10*n)/55. 11
0, 1, 123, 15128, 1860621, 228841255, 28145613744, 3461681649257, 425758697244867, 52364858079469384, 6440451785077489365, 792123204706451722511, 97424713727108484379488, 11982447665229637126954513, 1473743638109518258131025611, 181258485039805516112989195640 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Chebyshev polynomials S(n-1,123).
Used for all positive integer solutions of Pell equation x^2 - 5*(5*y)^2 = -4. See A097842 with A097843.
This is the k = 10 member of the k-family of sequences {F(k*n)/F(k)}, n >= 0 for k >= 1, with o.g.f. x/(1 - L(k)*x + (-1)^k*x^2). Proof: Binet-de Moivre formula for F and L. See also A028412. - Wolfdieter Lang, Aug 26 2012
LINKS
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Tanya Khovanova, Recursive Sequences
FORMULA
G.f.: x/(1-123*x+x^2), 123=L(10)=A000032(10) (Lucas).
a(n+1) = S(n, 123) = U(n, 123/2) = S(2*n+1, 5*sqrt(5))/(5*sqrt(5)), n>=0, with S(n, x) = U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x).
a(n) = 123*a(n-1) - a(n-2), n >= 2; a(0)=0, a(1)=1.
a(n) = (ap^n - am^n)/(ap-am) with ap := (123+55*sqrt(5))/2 and am := (123-55*sqrt(5))/2 = 1/ap.
From Peter Bala, Nov 29 2013: (Start)
a(n) = 1/(11*55)*(F(10*n + 5) - F(10*n - 5)).
Sum_{n >= 1} 1/( 11*a(n) + 1/(11*a(n)) ) = 1/11. Compare with A001906 and A049660. (End)
From Peter Bala, Apr 03 2015: (Start)
For integer k, 1 + k*(22 - k)*Sum_{n >= 1} a(n)*x^(2*n) = ( 1 + k/5*Sum_{n >= 1} Fibonacci(5*n)*x^n )*( 1 + k/5*Sum_{n >= 1} Fibonacci(5*n)*(-x)^n ).
1 + 4*Sum_{n >= 1} a(n)*x^(2*n) = ( 1 + 2/5*Sum_{n >= 1} Fibonacci(5*n+5)*x^n )*( 1 + 2/5*Sum_{n >= 1} Fibonacci(5*n+5)*(-x)^n ) = ( 1 + 2/5*Sum_{n >= 1} Fibonacci(5*n-5)*x^n )*( 1 + 2/5*Sum_{n >= 1} Fibonacci(5*n-5)*(-x)^n ).
1 + 25*Sum_{n >= 1} a(n)*x^(2*n) = ( 1 + Sum_{n >= 1} Fibonacci(5*n+3)*x^n )*( 1 + Sum_{n >= 1} Fibonacci(5*n+3)*(-x)^n ) = ( 1 + Sum_{n >= 1} Fibonacci(5*n-3)*x^n )*( 1 + Sum_{n >= 1} Fibonacci(5*n-3)*(-x)^n ).
1 + 100*Sum_{n >= 1} a(n)*x^(2*n) = ( 1 + 2*Sum_{n >= 1} Fibonacci(5*n+1)*x^n )*( 1 + 2*Sum_{n >= 1} Fibonacci(5*n+1)*(-x)^n ) = ( 1 + 2*Sum_{n >= 1} Fibonacci(5*n-1)*x^n )*( 1 + 2*Sum_{n >= 1} Fibonacci(5*n-1)*(-x)^n ).
1 + 125*Sum_{n >= 1} a(n)*x^(2*n) = ( 1 + Sum_{n >= 1} Lucas(5*n)*x^n )*( 1 + Sum_{n >= 1} Lucas(5*n)*(-x)^n ). (End)
MAPLE
seq(combinat:-fibonacci(10*n)/55, n=0..20); # Robert Israel, Apr 03 2015
MATHEMATICA
Table[Fibonacci[10 n]/55, {n, 12}] (* Michael De Vlieger, Apr 03 2015 *)
LinearRecurrence[{123, -1}, {0, 1}, 20] (* Harvey P. Dale, Dec 03 2019 *)
PROG
(MuPAD) numlib::fibonacci(10*n)/55 $ n = 0..25; // Zerinvary Lajos, May 09 2008
(PARI) a(n)=fibonacci(10*n)/55 \\ Charles R Greathouse IV, Oct 07 2016
(Magma) [ Fibonacci(10*n)/55: n in [0..30]]; // G. C. Greubel, Dec 02 2017
CROSSREFS
A column of array A028412.
Cf. A000045.
Sequence in context: A135479 A095761 A121917 * A181006 A362497 A289331
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Jan 20 2000
Chebyshev and Pell comments from Wolfdieter Lang, Sep 10 2004
STATUS
approved

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Last modified March 29 01:36 EDT 2024. Contains 371264 sequences. (Running on oeis4.)