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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
OFFSET
0,2
COMMENTS
See the comment section on A080144 for the general formula and the o.g.f. for b(m,n).
FORMULA
a(n) = b(6,n) = 4*A027941(n) + 9*A001654(n), with A027941(n) = Fibonacci(2*n+1) - 1 and A001654(n) = Fibonacci(n+1)*Fibonacci(n), n >= 0. 4 = Fibonacci(6)/2 and 9 = LucasL(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) = (1/10)*((29 - 13*sqrt(5))*((3 - sqrt(5))/2)^n + (29 + 13*sqrt(5))*((3 + sqrt(5))/2)^n - 2*(20 + 9*(-1)^n) ).
a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4) for n>3. (End)
a(n) = A001654(n+3) - 2*(2 + (-1)^n). - G. C. Greubel, Dec 31 2023
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 *)
LinearRecurrence[{3, 0, -3, 1}, {0, 13, 34, 102}, 40] (* Harvey P. Dale, Jun 13 2022 *)
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
(SageMath) [fibonacci(n+3)*fibonacci(n+4) - 2*(2+(-1)^n) for n in range(41)] # G. C. Greubel, Dec 31 2023
CROSSREFS
Cf. A001654, A027941, A059840(n+2), A064831, A080097, A080143 and A080144 for the m=0,1,...,5 members.
Cf. A027941.
Sequence in context: A069484 A089113 A067430 * A280322 A262851 A271750
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jul 27 2012
STATUS
approved