OFFSET
0,3
COMMENTS
Row sums of Riordan array (1,x*(1-5*x)). In general, a(n) = Sum_{k=0..n}(-1)^(n-k)*binomial(k,n-k)*r^(n-k), yields the row sums of the Riordan array (1,x*(1-k*x)).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..2859
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 16.
Index entries for linear recurrences with constant coefficients, signature (1,-5).
FORMULA
a(n) = ((1+sqrt(-19))^(n+1)-(1-sqrt(-19))^(n+1))/(2^(n+1)sqrt(-19)).
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k, n-k)*5^(n-k).
a(n) = 5^(n/2)(cos(-n*acot(sqrt(19)/19))-sqrt(19)sin(-n*acot(sqrt(19)/19))/19).
a(n) = a(n-1)-5*a(n-2), a(0)=1, a(1)=1. - Philippe Deléham, Oct 21 2008
a(n) = Sum_{k=0..n} A109466(n,k)*5^(n-k). - Philippe Deléham, Oct 25 2008
G.f.: Q(0)/2, where Q(k) = 1 + 1/( 1 - x*(2*k+1 -5*x)/( x*(2*k+2 -5*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 07 2013
MATHEMATICA
Join[{a=1, b=1}, Table[c=b-5*a; a=b; b=c, {n, 80}]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2011 *)
CoefficientList[Series[1/(1-x(1-5x)), {x, 0, 40}], x] (* or *) LinearRecurrence[ {1, -5}, {1, 1}, 40] (* Harvey P. Dale, Jan 21 2012 *)
PROG
(Sage) [lucas_number1(n, 1, 5) for n in range(1, 35)] # Zerinvary Lajos, Jul 16 2008
(PARI) Vec(1/(1-x+5*x^2) + O(x^99)) \\ Altug Alkan, Sep 06 2016
(Magma) I:=[1, 1]; [n le 2 select I[n] else Self(n-1) - 5*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 14 2018
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, May 08 2005
STATUS
approved