OFFSET
0,2
COMMENTS
This sequence does for Lucas numbers what A190173 does for Fibonacci numbers.
LINKS
Index entries for linear recurrences with constant coefficients, signature (4,-2,-6,4,2,-1).
FORMULA
The sums are (1) for L(2*k): (L(2*k+1)-1)^2 + L(2*k-1) + 1 and (2) for L(2*k+1): (L(2*k+2)-1)^2 + L(2*k) - 4.
G.f.: -x*(x^3+5*x^2-3*x-2) / ((x-1)*(x+1)*(x^2-3*x+1)*(x^2+x-1)). - Colin Barker, May 12 2014
a(n) = (L(n+1)-1)^2 + L(n-1) + (5*(-1)^n-3)/2. - Colin Barker, May 13 2014
EXAMPLE
For L(12) = a(13) the sum is (L(13)-1)^2 + L(11) + 1 = 520^2 + 200 = 270600 and for L(13) = a(14) the sum is (L(14)-1)^2 + l(12) - 4 = 842^2 + 322 - 4 = 709282.
PROG
(PARI) concat(0, Vec(-x*(x^3+5*x^2-3*x-2)/((x-1)*(x+1)*(x^2-3*x+1)*(x^2+x-1)) + O(x^100))) \\ Colin Barker, May 13 2014
(Sage)
[(lucas_number2(i+1, 1, -1)-1)^2+lucas_number2(i-1, 1, -1)+(5*(-1)^i-3)/2 for i in [0..50]] # Tom Edgar, May 13 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
J. M. Bergot, May 10 2014
EXTENSIONS
Typo in a(18) fixed by Colin Barker, May 12 2014
More terms from Colin Barker, May 12 2014
STATUS
approved