OFFSET
1,2
COMMENTS
p + q = 1, p*q = -5, p - q = sqrt(21).
The Lucas sequence V(1,-5) apart from the initial term a(0) = 2. - Peter Bala, Jun 23 2015
REFERENCES
Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", Wiley, 2001, p. 471.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (1,5).
FORMULA
G.f.: (10*x^2+x)/(1-x-5*x^2).
a(n) = n*sum(k=1..n, (C(k,n-k)*1^(2*k-n)*(5)^(n-k))/k). - Dmitry Kruchinin, May 16 2011
a(n) = a(n-1) + 5a(n-2), n>1.
a(n) = [x^n] ( (1 + x + sqrt(1 + 2*x + 21*x^2))/2 )^n. - Peter Bala, Jun 23 2015
EXAMPLE
a(5) = 151 = p^5 + q^5, with p = 2.79128...; q = -1.79128...
MATHEMATICA
CoefficientList[Series[(10 x + 1) / (1 - x - 5 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 20 2013 *)
PROG
(Sage) [lucas_number2(n, 1, -5) for n in range(1, 11)] # Zerinvary Lajos, May 14 2009
(Maxima) a(n):=n*sum((binomial(k, n-k)*1^(2*k-n)*(5)^(n-k))/k, k, 1, n) /* Dmitry Kruchinin, May 16 2011 */
(Magma) I:=[ 1, 11]; [n le 2 select I[n] else Self(n-1)+5*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jul 20 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Jul 02 2003
STATUS
approved