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A099279 Squares of A001076 (generalized Fibonacci). 7
0, 1, 16, 289, 5184, 93025, 1669264, 29953729, 537497856, 9645007681, 173072640400, 3105662519521, 55728852710976, 1000013686278049, 17944517500293904, 322001301319012225, 5778078906241926144 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

For the generalized Fibonacci sequences U(n-1;a):=(ap(a)^n - am(a)^n)/(ap(a)-am(a)) with ap(a):=(a+sqrt(a^2+4))/2, am(a):=(a-sqrt(a^2+4))/2, a from the integers, one has for the squared sequences U(n-1;a)^2 = (2*T(n,(a^2+2)/2) - 2*(-1)^n)/(a^2+4). Here T(n,x) are Chebyshev's polynomials of the first kind (see A053120). Therefore the o.g.f. for the squared sequence is x*(1-x)/((1-(a^2+2)*x+x^2)*(1+x)) = x*(1-x)/(1-(a^2+1)*x-(a^2+1)*x^2+x^3). For this example a=4.

Unsigned member r=-16 of the family of Chebyshev sequences S_r(n) defined in A092184.

((-1)^(n+1))*a(n) = S_{-16}(n), n>=0, defined in A092184.

LINKS

Table of n, a(n) for n=0..16.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (17, 17, -1).

FORMULA

a(n) = A001076(n)^2.

a(n) = 17*a(n-1) + 17*a(n-2) - a(n-3), n >= 3; a(0)=0, a(1)=1, a(2)=16.

a(n) = 18*a(n-1) - a(n-2) - 2*(-1)^n, n > =2; a(0)=0, a(1)=1.

a(n) = (T(n, 9) - (-1)^n)/10 with Chebyshev's T(n, x) polynomials of the first kind. T(n, 9)=A023039(n).

G.f.: x*(1-x)/((1-18*x+x^2)*(1+x)) = x*(1-x)/(1-17*x-17*x^2+x^3).

a(n) = -(1/10)*(-1)^n + (1/20)*(9-4*sqrt(5))^n + (1/20)*(9+4*sqrt(5))^n, with n >= 0. - Paolo P. Lava, Aug 27 2008

a(n) = a(n-1) + A001654(3*n-2) with a(0)=0, where A001654 are the golden rectangle numbers. - Johannes W. Meijer, Sep 22 2010

MAPLE

with (combinat):seq(fibonacci(n, 4)^2, n=0..16); # Zerinvary Lajos, Apr 09 2008

nmax:=48: with(combinat): for n from 0 to nmax do A001654(n):=fibonacci(n) * fibonacci(n+1) od: a(0):=0: for n from 1 to nmax/3 do a(n):=a(n-1)+A001654(3*n-2) od: seq(a(n), n=0..nmax/3); # Johannes W. Meijer, Sep 22 2010

MATHEMATICA

LinearRecurrence[{17, 17, -1}, {0, 1, 16}, 30] (* Harvey P. Dale, Mar 26 2012 *)

PROG

(MuPAD) numlib::fibonacci(3*n)^2/4 $ n = 0..35; // Zerinvary Lajos, May 13 2008

(Sage) [(fibonacci(3*n))^2/4 for n in range(0, 17)] # Zerinvary Lajos, May 15 2009

(PARI) x='x+O('x^99); concat([0], Vec(x*(1-x)/((1-18*x+x^2)*(1+x)))) \\ Altug Alkan, Dec 17 2017

CROSSREFS

Cf. A007598, A079291, A092936, A099365-6 (other square sequences of this type).

Sequence in context: A225194 A027776 A140770 * A202878 A183886 A230341

Adjacent sequences:  A099276 A099277 A099278 * A099280 A099281 A099282

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Oct 18 2004

STATUS

approved

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Last modified December 3 18:19 EST 2021. Contains 349467 sequences. (Running on oeis4.)