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A214729 Member m=6 of the m-family of sums b(m,n) = Sum_{k=0..n} F(k+m)*F(k), m >= 0, n >= 0, with the Fibonacci numbers F. 1
0, 13, 34, 102, 267, 712, 1864, 4893, 12810, 33550, 87835, 229968, 602064, 1576237, 4126642, 10803702, 28284459, 74049688, 193864600, 507544125, 1328767770, 3478759198, 9107509819, 23843770272, 62423800992, 163427632717, 427859097154, 1120149658758 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

See the comment section on A080144 for the general formula and the o.g.f. for b(m,n).

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

FORMULA

a(n) = b(6,n) = 4*A027941(n) + 9*A001654(n), with A027941(n) = F(2*n+1)-1 and A001654(n) = F(n+1)*F(n), n >= 0. 4 = F(6)/2 and 9 = L(6)/2.

O.g.f.: x*(13-5*x)/((1-x^2)*(1-3*x+x^2)) (see a comment above). - Wolfdieter Lang, Jul 30 2012

a(n) = (9*(-1)^(n+1) - 20 + Lucas(2*n + 7))/5. - Ehren Metcalfe, Aug 21 2017

From Colin Barker, Aug 25 2017: (Start)

a(n) =  (2^(-1-n)*(-2^(1+n)*(20 + 9*(-1)^n) + (29-13*sqrt(5))*(3-sqrt(5))^n + (3+sqrt(5))^n*(29+13*sqrt(5)))) / 5.

a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4) for n>3.

(End)

MATHEMATICA

With[{m = 6}, Table[Sum[Fibonacci[k + m]*Fibonacci[k], {k, 0, n}], {n, 0, 25}]] (* or *)

Table[(9 (-1)^(n + 1) - 20 + LucasL[2 n + 7])/5, {n, 0, 25}] (* Michael De Vlieger, Aug 23 2017 *)

PROG

(PARI) concat(0, Vec(x*(13 - 5*x) / ((1 - x)*(1 + x)*(1 - 3*x + x^2)) + O(x^30))) \\ Colin Barker, Aug 25 2017

(MAGMA) [(9*(-1)^(n+1)-20+Lucas(2*n+7))/5: n in [0..40]]; // Vincenzo Librandi, Aug 26 2017

CROSSREFS

Cf. A001654, A064831, A059840(n+2), A080097, A080143 and A080144 for the m=0,1,...,5 members.

Sequence in context: A069484 A089113 A067430 * A280322 A262851 A271750

Adjacent sequences:  A214726 A214727 A214728 * A214730 A214731 A214732

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Jul 27 2012

STATUS

approved

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Last modified February 20 13:10 EST 2020. Contains 332076 sequences. (Running on oeis4.)