OFFSET
0,2
COMMENTS
The present sequence is a member of the m-family of sums b(m,n) := Sum_{k=1..n} F(k+m)*F(k) for m >= 0, n >= 0 (see the B. Cloitre formula given below (m=5)). These sums are (F(m)*A027941(n) + L(m)*A001654(n))/2, with F = A000045 and the L = A000032. Proof by induction on m using the recurrence. - Wolfdieter Lang, Jul 27 2012
The o.g.f. of b(m,n) is A(m,x) = x*(F(m+1) - F(m-1)*x)/((1-x^2)*(1-3*x+x^2)), m >= 0, with F(-1)=1. - Wolfdieter Lang, Jul 30 2012
b(m,n) = ((-1)^(n+1)*L(m) - 5*F(m) + 2*L(m + 2*n + 1))/10. - Ehren Metcalfe, Aug 21 2017
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
S. Falcon, On the Sequences of Products of Two k-Fibonacci Numbers, American Review of Mathematics and Statistics, March 2014, Vol. 2, No. 1, pp. 111-120.
FORMULA
G.f.: x*(8-3*x)/((1-x^2)*(1-3*x+x^2)) (see the comment section). - Wolfdieter Lang, Jul 30 2012
a(n) = (5*A027941(n) + 11*A001654(n))/2, n >= 0. See A080143 and A080097. See the comment section for the general formula for such sums. - Wolfdieter Lang, Jul 27 2012
a(n) = (2*Lucas(2*n + 6) + 11*(-1)^(n+1) - 25)/10. - Ehren Metcalfe, Aug 21 2017
a(n) = (2*Fibonacci(n+3)^2 - 5 - 3*(-1)^n)/2. - G. C. Greubel, Jul 23 2019
MATHEMATICA
CoefficientList[Series[x*(8+5*x-3*x^2)/((1-x^2)*(1-2x-2x^2+x^3)), {x, 0, 30}], x]
With[{F=Fibonacci}, Table[(2*F[n + 3]^2 -5-3*(-1)^n)/2, {n, 0, 30}]] (* G. C. Greubel, Jul 23 2019 *)
PROG
(PARI) my(x='x+O('x^30)); concat([0], Vec(x*(8-3*x)/((1-x^2)*(1-3*x+x^2)) )) \\ G. C. Greubel, Mar 04 2017
(PARI) vector(30, n, n--; f=fibonacci; (2*f(n+3)^2 -5-3*(-1)^n)/2) \\ G. C. Greubel, Jul 23 2019
(Magma) F:=Fibonacci; [(2*F(n+3)^2 -5-3*(-1)^n)/2: n in [0..30]]; // G. C. Greubel, Jul 23 2019
(Sage) f=fibonacci; [(2*f(n+3)^2 -5-3*(-1)^n)/2 for n in (0..30)] # G. C. Greubel, Jul 23 2019
(GAP) F:=Fibonacci;; List([0..30], n-> (2*F(n+3)^2 -5-3*(-1)^n)/2); # G. C. Greubel, Jul 23 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Mario Catalani (mario.catalani(AT)unito.it), Jan 31 2003
STATUS
approved