OFFSET
0,2
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,-1).
FORMULA
a(n) = (8*(((3 + sqrt(5))/2)^n - ((3 - sqrt(5))/2)^n) - (((3 + sqrt(5))/2)^(n - 1) - ((3 - sqrt(5))/2)^(n - 1)))/sqrt(5).
G.f.: (1 + 5*x)/(1 - 3*x + x^2).
From Rigoberto Florez, Dec 24 2018: (Start)
a(n) = Fibonacci(2n+2) + 5*Fibonacci(2n),
a(n) = 3*Fibonacci(2n+2) - Fibonacci(2n-3). (End)
E.g.f.: (1/5)*exp(3*x/2)*(5*cosh(sqrt(5)*x/2) + 13*sqrt(5)*sinh(sqrt(5)*x/2)). - Franck Maminirina Ramaharo, Dec 26 2018
a(n) = ChebyshevT(n, 3/2) + (13/2)*ChebyshevU(n-1, 3/2) = ChebyshevU(n, 3/2) + 5*ChebyshevU(n-1, 3/2). - G. C. Greubel, Jan 17 2020
MAPLE
seq(coeff(series((1+5*x)/(1-3*x+x^2), x, n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Dec 29 2018
MATHEMATICA
Table[3Fibonacci[2n+2]-Fibonacci[2n-3], {n, 0, 20}] (* Rigoberto Florez, Dec 24 2018 *)
LinearRecurrence[{3, -1}, {1, 8}, 30] (* Vincenzo Librandi, Dec 25 2018 *)
PROG
(Magma) [Fibonacci(2*n+2) + 5*Fibonacci(2*n): n in [0..30]]; // Vincenzo Librandi, Dec 25 2018
(GAP) a:=[1, 8];; for n in [3..30] do a[n]:=3*a[n-1]-a[n-2]; od; Print(a); # Muniru A Asiru, Dec 29 2018
(PARI) vector(30, n, fibonacci(2*n) + 5*fibonacci(2*n-2) ) \\ G. C. Greubel, Jan 17 2020
(Sage) [fibonacci(2*n+2) +5*fibonacci(2*n) for n in (0..30)] # G. C. Greubel, Jan 17 2020
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, May 28 2000
STATUS
approved