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A079291 Squares of Pell numbers. 20
0, 1, 4, 25, 144, 841, 4900, 28561, 166464, 970225, 5654884, 32959081, 192099600, 1119638521, 6525731524, 38034750625, 221682772224, 1292061882721, 7530688524100, 43892069261881, 255821727047184, 1491038293021225 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n)*(-1)^(n+1) is the r=-4 member of the r-family of sequences S_r(n), n>=1, defined in A092184 where more information can be found.

Binomial transform of A086346. - Johannes W. Meijer, Aug 01 2010

LINKS

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

Joerg Arndt, Matters Computational (The Fxtbook), sect. 32.1.5, pp. 626-627

T. Mansour, A note on sum of k-th power of Horadam's sequence

T. Mansour, Squaring the terms of an ell-th order linear recurrence

P. Stanica, Generating functions, weighted and non-weighted sums of powers...

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (5,5,-1).

FORMULA

G.f.: x*(1-x)/(1+x)/(1-6x+x^2).

a(n) = (r^n+(1/r)^n-2*(-1)^n)/8, with r=3+sqrt(8).

a(n+3) = 5*a(n+2)+5*a(n+1)-a(n).

L.g.f.: 1/8*log((1+2*x+x^2)/(1-6*x+x^2)) = sum(n>=0, a(n)/n*x^n), see p.627 of the Fxtbook link; special case of the following: let v(0)=0, v(1)=1, and v(n)=u*v(n-1)+v(n-2), then 1/A*log((1+2*x+x^2)/(1-(2-A)*x+x^2)) = sum(n>=0, v(n)^2/n*x^n) where A=u^2+4.  [Joerg Arndt, Apr 08 2011]

a(n+1) = sum_{k=0...n}((-1)^(n-k)*A001653(k)); e.g. 144 = -1 + 5 - 29 + 169; 25 = 1 - 5 + 29 - Charlie Marion, Jul 16 2003

a(n)=A000129(n)^2.

a(n)= (T(n, 3)-(-1)^n)/4 with Chebyshev's polynomials of the first kind evaluated at x=3: T(n, 3)=A001541(n)=((3+2*sqrt(2))^n + (3-2*sqrt(2))^n)/2. - Wolfdieter Lang, Oct 18 2004

a(n) = rightmost term of M^n * [1 0 0] where M = the 3 X 3 matrix [4 4 1 / 2 1 0 / 1 0 0]. a(n+1) = leftmost term. E.g. a(6) = 4900, a(5) = 841 since M^5 * [1 0 0] = [4900 2030 841]. - Gary W. Adamson, Oct 31 2004

a(n) = [(-1)^(n+1)+A001109(n+1)-3*A001109(n)]/4. - R. J. Mathar, Nov 16 2007

a(n) = ( ( ( (1-Sqrt[ 2 ])^n + (1+Sqrt[ 2 ])^n) /2 )^2 + (-1)^(n+1) ) /2 - Antonio Pane (apane1(AT)spc.edu), Dec 15 2007

Limit(a(n+k)/a(k),k=infinity) = A001541(n)+2*A001109(n)*sqrt(2). - Johannes W. Meijer, Aug 01 2010

For n>0, a(2n) = 6*a(2n-1)-a(2n-2)-2, a(2n+1) = 6*a(2n)-a(2n-1)+2. - Charlie Marion, Sep 24 2011

MAPLE

with(combinat):seq(fibonacci(i, 2)^2, i=0..21); # Zerinvary Lajos, Mar 20 2008

MATHEMATICA

CoefficientList[Series[x (1 - x) / (1 + x) / (1 - 6 x + x^2), {x, 0, 30}], x] (* Vincenzo Librandi, May 17 2013 *)

LinearRecurrence[{5, 5, -1}, {0, 1, 4}, 40] (* Harvey P. Dale, Dec 20 2015 *)

PROG

(MAGMA) I:=[0, 1, 4]; [n le 3 select I[n] else 5*Self(n-1)+ 5*Self(n-2) - Self(n-3): n in [1..25]]; // Vincenzo Librandi, May 17 2013

CROSSREFS

Cf. A000129, A001254, A007598, A084158.

Sequence in context: A278275 A301836 A275177 * A173612 A072221 A302064

Adjacent sequences:  A079288 A079289 A079290 * A079292 A079293 A079294

KEYWORD

easy,nonn

AUTHOR

Ralf Stephan, Feb 08 2003

STATUS

approved

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Last modified August 10 16:56 EDT 2020. Contains 336381 sequences. (Running on oeis4.)