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A056854 a(n) = Lucas(4*n). 16
2, 7, 47, 322, 2207, 15127, 103682, 710647, 4870847, 33385282, 228826127, 1568397607, 10749957122, 73681302247, 505019158607, 3461452808002, 23725150497407, 162614600673847, 1114577054219522, 7639424778862807, 52361396397820127, 358890350005878082, 2459871053643326447 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n) and b(n) := A004187(n) are the nonnegative proper and improper solutions of the Pell equation a(n)^2 - 5*(3*b(n))^2 = +4. See the cross-reference to A004187 below. - Wolfdieter Lang, Jun 26 2013

Lucas numbers of the form n^2-2. - Michel Lagneau, Aug 11 2014

REFERENCES

E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pps. 231-242.

D. Gerdemann, Combinatorial proofs of Zeckendorf family identities, Fib. Q., 46/47 (2008/2009), 249-261. [From N. J. A. Sloane, Dec 05 2009]

A. F. Horadam, Special Properties of the Sequence W(n){a,b; p,q}, Fib. Quart., 5 (1967), pps. 424-434.

LINKS

Table of n, a(n) for n=0..22.

P. Bala, Some simple continued fraction expansions for an infinite product, Part 1

Tanya Khovanova, Recursive Sequences

Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (7,-1)

FORMULA

a(n) = 7*a(n-1) - a(n-2) with a(0)=2, a(1)=7.

a(n) = A000032(4*n), where A000032 = Lucas numbers.

a(n) = 7*S(n-1, 7) - 2*S(n-2, 7) = S(n, 7) - S(n-2, 7) = 2*T(n, 7/2), with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. U(n, x), resp. T(n, x), are Chebyshev's polynomials of the second, resp. first, kind. S(n-1, 7) = A004187(n), n>=0. See A049310 and A053120.

a(n) = ((7+sqrt(45))/2)^n + ((7-sqrt(45))/2)^n.

G.f.: (2-7x)/(1-7x+x^2).

a(n) = A005248(2*n); bisection of A005248.

a(n) = Fibonacci(8*n)/Fibonacci(4*n), n>0. - Gary Detlefs Dec 26 2010

a(n) = 2 + 5*Fibonacci(2*n)^2 = 2 + 5*A049684(n), n >= 0. This is in Koshy's book (reference under A065563) 15. on p. 88. Compare with the above Chebyshev T formula. - Wolfdieter Lang, Aug 27 2012

From Peter Bala, Jan 06 2013: (Start)

Let F(x) = product {n = 0..inf} (1 + x^(4*n+1))/(1 + x^(4*n+3)). Let alpha = 1/2*(7 - 3*sqrt(5)). This sequence gives the simple continued fraction expansion of 1 + F(alpha) = 2.14242 42709 40138 85949 ... = 2 + 1/(7 + 1/(47 + 1/(322 + ...))).

Also F(-alpha) = 0.85670 72882 04563 14901 ... has the continued fraction representation 1 - 1/(7 - 1/(47 - 1/(322 - ...))) and the simple continued fraction expansion 1/(1 + 1/((7-2) + 1/(1 + 1/((47-2) + 1/(1 + 1/((322-2) + 1/(1 + ...))))))). Cf. A005248.

F(alpha)*F(-alpha) has the simple continued fraction expansion 1/(1 + 1/((7^2-4) + 1/(1 + 1/((47^2-4) + 1/(1 + 1/((322^2-4) + 1/(1 + ...))))))).

(End)

a(n) = Fibonacci(4*n+2) - Fibonacci(4*n-2), where Fibonacci(-2) = -1. [Bruno Berselli, May 25 2015]

EXAMPLE

Pell equation: n = 0, 2^2 - 45*0^2 = +4 (improper);  n = 1, 7^2 - 5*(3*1)^2 = +4; n=2, 47^2 - 5*(3*7)^2 = +4. - Wolfdieter Lang, Jun 26 2013

MATHEMATICA

a[0] = 2; a[1] = 7; a[n_] := 7a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 19}] (* Robert G. Wilson v, Jan 30 2004 *)

LinearRecurrence[{7, -1}, {2, 7}, 25] (* or *) LucasL[4*Range[0, 25]] (* Harvey P. Dale, Aug 08 2011 *)

PROG

(PARI) a(n)=if(n<0, 0, polsym(1-7*x+x^2, n)[n+1])

(PARI) a(n)=if(n<0, 0, 2*subst(poltchebi(n), x, 7/2))

(Sage) [lucas_number2(n, 7, 1) for n in range(27)] - Zerinvary Lajos, Jun 25 2008

(MAGMA) [Lucas(4*n): n in [0..100]]; // Vincenzo Librandi, Apr 14 2011

CROSSREFS

a(n)=sqrt{[45*(A004187(n))^2]+4}. A005248.

Cf. quadrisection of A000032: this sequence (first), A056914 (second), A246453 (third, without 11), A288913 (fourth).

Sequence in context: A072287 A276649 A091117 * A117141 A125813 A254439

Adjacent sequences:  A056851 A056852 A056853 * A056855 A056856 A056857

KEYWORD

nonn,easy,changed

AUTHOR

Barry E. Williams, Aug 29 2000

EXTENSIONS

More terms from James A. Sellers, Aug 31 2000

Chebyshev comments from Wolfdieter Lang, Oct 31 2002

STATUS

approved

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Last modified June 22 08:03 EDT 2017. Contains 288605 sequences.