OFFSET
1,3
COMMENTS
See A075271 for the definition of the BinomialMean transform.
The inverse binomial transform of 2^n*c(n+1), where c(n) is the solution to c(n) = c(n-1) + k*c(n-2), a(0)=0, a(1)=1 is 1, 1, 4k+1, 4k+1, (4k+1)^2, ... - Paul Barry, Feb 12 2004
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (0,5).
FORMULA
a(n) = 5^floor((n-1)/2).
a(1)=1, a(2)=1 and, for n > 2, a(n) = 5*a(n-2).
From Paul Barry, Feb 12 2004: (Start)
G.f.: x*(1+x)/(1-5*x^2);
a(n) = (1/(2*sqrt(5))*((1+sqrt(5))*(sqrt(5))^n - (1-sqrt(5))*(-sqrt(5))^n)).
Inverse binomial transform of A063727 (2^n*Fibonacci(n+1)). (End)
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
E.g.f.: (cosh(sqrt(5)*x) + sqrt(5)*sinh(sqrt(5)*x) - 1)/5. - Stefano Spezia, May 24 2024
MATHEMATICA
a[1] := 1; a[2] := 1; a[n_] := 5a[n - 2]; Table[a[n], {n, 30}] (* Alonso del Arte, Mar 04 2011 *)
PROG
(Magma) [5^Floor((n-1)/2): n in [1..40]]; // Vincenzo Librandi, Aug 16 2011
(PARI) a(n)=5^((n-1)\2) \\ Charles R Greathouse IV, Oct 03 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
John W. Layman, Sep 12 2002
STATUS
approved