OFFSET
0,3
COMMENTS
This is a divisibility sequence.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..595
Index entries for linear recurrences with constant coefficients, signature (48,-48,1).
FORMULA
a(n) = (1/45)*(-2 + ((47 + 7*sqrt(45))/2)^n + ((47 - 7*sqrt(45))/2)^n). - Ralf Stephan, Apr 14 2004
From R. J. Mathar, Jun 03 2009: (Start)
a(n) = (A004187(n))^2.
a(n) = 48*a(n-1) - 48*a(n-2) + a(n-3).
G.f.: x*(1 + x)/((1 - x)*(1 - 47*x + x^2)). (End)
From R. K. Guy, Feb 24 2010: (Start)
a(n) = F(4*n)^2/9.
a(n) - a(n-1) = A004187(2n-1). (End)
From Peter Bala, Jun 03 2016: (Start)
exp( Sum_{n >= 1} 45*a(n)*x^n/n ) = 1 + 15/7*Sum_{n >= 1} Fibonacci(8*n)*x^n.
This is the particular case k = 4 of the relation exp( Sum_{n >= 1} 5*F(k*n)^2*x^n/n ) = 1 + 5*Fibonacci(k)/Lucas(k) * ( Sum_{n >= 1} F(2*k*n)*x^n ). (End)
Lim_{n->infinity} a(n+1)/a(n) = (47 + 21*sqrt(5))/2 = phi^8, where phi is the golden ratio (A001622). - Ilya Gutkovskiy, Jun 06 2016
a(n) = a(-n) for all n in Z. - Michael Somos, Jun 12 2016
0 = a(n)*(+a(n) -98*a(n+1) -2*a(n+2)) + a(n+1)*(+2401*a(n+1) -98*a(n+2)) + a(n+2)^2 for all integer n. - Michael Somos, Jun 12 2016
EXAMPLE
G.f. = x + 49*x^2 + 2304*x^3 + 108241*x^4 + 5085025*x^5 + 238887936*x^6 + ...
MAPLE
with(combinat); seq( fibonacci(4*n)^2/9, n=0..20); # G. C. Greubel, Dec 14 2019
MATHEMATICA
LinearRecurrence[{48, -48, 1}, {0, 1, 49}, 20] (* or *) CoefficientList[Series[ (-x-x^2)/ (x^3-48x^2+48x-1), {x, 0, 20}], x] (* Harvey P. Dale, Apr 22 2011 *)
PROG
(MuPAD) numlib::fibonacci(4*n)^2/9 $ n = 0..25; // Zerinvary Lajos, May 09 2008
(PARI) vector(21, n, (fibonacci(4*(n-1))/3)^2) \\ G. C. Greubel, Dec 02 2017
(Magma) [(Fibonacci(4*n)/3)^2: n in [0..20]]; // G. C. Greubel, Dec 02 2017
(Sage) [(fibonacci(4*n)/3)^2 for n in (0..20)] # G. C. Greubel, Dec 14 2019
(GAP) List([0..20], n-> (Fibonacci(4*n)/3)^2 ); # G. C. Greubel, Dec 14 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from N. J. A. Sloane, Feb 26 2010
STATUS
approved