|
| |
|
|
A099279
|
|
Squares of A001076 (generalized Fibonacci).
|
|
6
| |
|
|
0, 1, 16, 289, 5184, 93025, 1669264, 29953729, 537497856, 9645007681, 173072640400, 3105662519521, 55728852710976, 1000013686278049, 17944517500293904, 322001301319012225, 5778078906241926144
(list; graph; refs; listen; history; 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
| Index entries for sequences related to Chebyshev polynomials.
|
|
|
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 [From Paolo P. Lava (paoloplava(AT)gmail.com), Aug 27 2008]
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Sep 22 2010: (Start)
a(n) = a(n-1)+A001654(3*n-2) with a(0)=0. The A001654(n) are the golden rectangle numbers.
(End)
|
|
|
MAPLE
| with (combinat):seq(fibonacci(n, 4)^2, n=0..16); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 09 2008
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Sep 22 2010: (Start)
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);
(End)
|
|
|
PROG
| (Mupad) numlib::fibonacci(3*n)^2/4 $ n = 0..35; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 13 2008
(Other) sage: [(fibonacci(3*n))^2/4 for n in xrange(0, 17)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009]
|
|
|
CROSSREFS
| Cf. A007598, A079291, A092936, A099365-6 (other square sequences of this type).
Sequence in context: A182608 A027776 A140770 * A202878 A183886 A039746
Adjacent sequences: A099276 A099277 A099278 * A099280 A099281 A099282
|
|
|
KEYWORD
| nonn,easy
|
|
|
AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 18 2004
|
| |
|
|
