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A080144 a(n) = F(4)*F(n)*F(n+1) + F(5)*F(n+1)^2 if n odd, a(n) = F(4)*F(n)*F(n+1) + F(5)*F(n+1)^2 - F(5) if n even, where F(n) is the n-th Fibonacci number (A000045). 10
0, 8, 21, 63, 165, 440, 1152, 3024, 7917, 20735, 54285, 142128, 372096, 974168, 2550405, 6677055, 17480757, 45765224, 119814912, 313679520, 821223645, 2149991423, 5628750621, 14736260448, 38580030720, 101003831720 (list; graph; refs; listen; history; text; internal format)
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) = Sum_{i=0..n} A000045(i+5)*A000045(i). - Benoit Cloitre, Jun 14 2004

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

Cf. A000032, A000045, A059840, A064831, A080143.

Sequence in context: A296198 A301538 A192299 * A241522 A096018 A297647

Adjacent sequences:  A080141 A080142 A080143 * A080145 A080146 A080147

KEYWORD

nonn,easy

AUTHOR

Mario Catalani (mario.catalani(AT)unito.it), Jan 31 2003

STATUS

approved

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Last modified April 7 13:36 EDT 2020. Contains 333305 sequences. (Running on oeis4.)