|
| |
|
|
A049682
|
|
a(n) = (Lucas(8*n) - 2)/45.
|
|
7
|
|
|
|
0, 1, 49, 2304, 108241, 5085025, 238887936, 11222647969, 527225566609, 24768378982656, 1163586586618225, 54663801192073921, 2568035069440856064, 120642984462528161089, 5667652234669382715121, 266259012044998459449600, 12508505913880258211416081
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
This is a divisibility sequence.
|
|
|
LINKS
|
G. C. Greubel, Table of n, a(n) for n = 0..595
Index to divisibility sequences.
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
|
Cf. A000032, A000045, A004146, A049683, A049684.
Sequence in context: A163927 A245036 A061615 * A162914 A163287 A163835
Adjacent sequences: A049679 A049680 A049681 * A049683 A049684 A049685
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
Clark Kimberling
|
|
|
EXTENSIONS
|
More terms from N. J. A. Sloane, Feb 26 2010
|
|
|
STATUS
|
approved
|
| |
|
|
