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A081068 a(n) = (Lucas(4*n+2) + 2)/5, or Fibonacci(2*n+1)^2, or A081067(n)/5. 10
1, 4, 25, 169, 1156, 7921, 54289, 372100, 2550409, 17480761, 119814916, 821223649, 5628750625, 38580030724, 264431464441, 1812440220361, 12422650078084, 85146110326225, 583600122205489, 4000054745112196, 27416783093579881 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

First differences of A103433.

REFERENCES

Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 19.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876 (See Corollary 1 (vii)).

Giovanni Lucca, Integer Sequences and Circle Chains Inside a Circular Segment, Forum Geometricorum, Vol. 18 (2018), 47-55.

Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

FORMULA

Equals A001519(n)^2 and A058038 - 1.

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).

a(n) = (2/5)+(3/10)*{[(7/2)-(3/2)*sqrt(5)]^n+[(7/2)+(3/2)*sqrt(5)]^n}+(1/10)*sqrt(5)*{[(7/2) +(3/2)*sqrt(5)]^n-[(7/2)-(3/2)*sqrt(5)]^n}, with n>=0. - Paolo P. Lava, Dec 01 2008

a(n) = Fibonacci(2*n)*Fibonacci(2*n+2) +1. - Gary Detlefs, Apr 01 2012

G.f.: (1-4*x+x^2)/((1-x)*(x^2-7*x+1)). - Colin Barker, Jun 26 2012

Sum {n >= 0} 1/(a(n) + 1) = 1/3*sqrt(5). - Peter Bala, Nov 30 2013

MAPLE

luc := proc(n) option remember: if n=0 then RETURN(2) fi: if n=1 then RETURN(1) fi: luc(n-1)+luc(n-2): end: for n from 0 to 40 do printf(`%d, `, (luc(4*n+2)+2)/5) od: # James A. Sellers, Mar 05 2003

MATHEMATICA

CoefficientList[Series[-(1-4*x+x^2)/((x-1)*(x^2-7*x+1)), {x, 0, 40}], x] (* or *) LinearRecurrence[{8, -8, 1}, {1, 4, 25}, 50] (* Vincenzo Librandi, Jun 26 2012 *)

Table[(LucasL[4*n+2] + 2)/5, {n, 0, 30}] (* G. C. Greubel, Dec 17 2017 *)

PROG

(MAGMA)  I:=[1, 4, 25]; [n le 3 select I[n] else 8*Self(n-1)-8*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 26 2012

(PARI) main(size)={ return(concat([1], vector(size, n, fibonacci(2*n+1)^2))) } /* Anders Hellström, Jul 11 2015 */

(MAGMA) [(Lucas(4*n+2) + 2)/5: n in [0..30]]; // G. C. Greubel, Dec 17 2017

(PARI) for(n=0, 30, print1(fibonacci(2*n+1)^2, ", ")) \\ G. C. Greubel, Dec 17 2017

CROSSREFS

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A081067.

Sequence in context: A006880 A227693 A175255 * A163072 A278689 A140177

Adjacent sequences:  A081065 A081066 A081067 * A081069 A081070 A081071

KEYWORD

nonn,easy

AUTHOR

R. K. Guy, Mar 04 2003

EXTENSIONS

More terms from James A. Sellers, Mar 05 2003

STATUS

approved

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Last modified April 24 20:29 EDT 2019. Contains 322446 sequences. (Running on oeis4.)