A049670 - OEIS

A049670

a(n) = Fibonacci(10*n)/55.

0, 1, 123, 15128, 1860621, 228841255, 28145613744, 3461681649257, 425758697244867, 52364858079469384, 6440451785077489365, 792123204706451722511, 97424713727108484379488, 11982447665229637126954513, 1473743638109518258131025611, 181258485039805516112989195640

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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

Robert Israel, Table of n, a(n) for n = 0..383

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

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (123,-1).

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.