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A049684 a(n) = Fibonacci(2n)^2. 20
0, 1, 9, 64, 441, 3025, 20736, 142129, 974169, 6677056, 45765225, 313679521, 2149991424, 14736260449, 101003831721, 692290561600, 4745030099481, 32522920134769, 222915410843904, 1527884955772561, 10472279279564025 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

This is the r=9 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found.

REFERENCES

Mohammad K. Azarian, Fibonacci Identities as Binomial Sums II, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 42, 2012, pp. 2053-2059.

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

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 27.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.

E. Kilic, Y. T. Ulutas, N. Omur, A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters, J. Int. Seq. 14 (2011) #11.5.6, table 1, k=2.

R. Stephan, Boring proof of a nonlinearity

Index entries for sequences related to Chebyshev polynomials.

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

FORMULA

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

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3) with n>2, a(0)=0, a(1)=1, a(2)=9.

a(n) = 7*a(n-1) - a(n-2) + 2 = A001906(n)^2.

a(n) = (A000032(4*n)-2)/5. [This is in Koshy's book (reference under A065563) on p. 88, attributed to Lucas 1876.] - Wolfdieter Lang, Aug 27 2012]

a(n) = 1/5*(-2 + ( (7+sqrt(45))/2 )^n + ( (7-sqrt(45))/2 )^n). - Ralf Stephan, Apr 14 2004

a(n)= 2*(T(n, 7/2)-1)/5 with twice the Chebyshev's polynomials of the first kind evaluated at x=7/2: 2*T(n, 7/2)= A056854(n). - Wolfdieter Lang, Oct 18 2004

a(n) = F(n-1)*F(n+1)-1. [Bruno Berselli, Feb 12 2015]

a(n) = Sum_{i=1..n} F(4*i-2) for n>0. [Bruno Berselli, Aug 25 2015]

MATHEMATICA

Join[{a=0, b=1}, Table[c=7*b-1*a+2; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)

Fibonacci[Range[0, 40, 2]]^2 (* Harvey P. Dale, Mar 22 2012 *)

Table[Fibonacci[n - 1] Fibonacci[n + 1] - 1, {n, 0, 40, 2}] (* Bruno Berselli, Feb 12 2015 *)

LinearRecurrence[{8, -8, 1}, {0, 1, 9}, 21] (* Ray Chandler, Sep 23 2015 *)

PROG

(PARI) a(n)=fibonacci(2*n)^2

(MuPAD) numlib::fibonacci(2*n)^2 $ n = 0..35; // Zerinvary Lajos, May 13 2008

(Sage) [fibonacci(2*n)^2 for n in xrange(0, 21)] # Zerinvary Lajos, May 15 2009

CROSSREFS

First differences give A033890.

First differences of A103434.

Bisection of A007598 and A064841.

a(n) = A064170(n+2) - 1 = (1/5) A081070.

Cf. A000045.

Sequence in context: A083328 A000846 A231822 * A037540 A037484 A013566

Adjacent sequences:  A049681 A049682 A049683 * A049685 A049686 A049687

KEYWORD

nonn,nice,easy

AUTHOR

Clark Kimberling

EXTENSIONS

Better description and more terms from Michael Somos

STATUS

approved

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Last modified January 18 20:19 EST 2018. Contains 297865 sequences.