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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.

%I A214729 #37 Dec 31 2023 03:41:55
%S A214729 0,13,34,102,267,712,1864,4893,12810,33550,87835,229968,602064,
%T A214729 1576237,4126642,10803702,28284459,74049688,193864600,507544125,
%U A214729 1328767770,3478759198,9107509819,23843770272,62423800992,163427632717,427859097154,1120149658758
%N 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.
%C A214729 See the comment section on A080144 for the general formula and the o.g.f. for b(m,n).
%H A214729 Colin Barker, <a href="/A214729/b214729.txt">Table of n, a(n) for n = 0..1000</a>
%H A214729 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-3,1).
%F A214729 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.
%F A214729 O.g.f.: x*(13-5*x)/((1-x^2)*(1-3*x+x^2)) (see a comment above). - _Wolfdieter Lang_, Jul 30 2012
%F A214729 a(n) = (9*(-1)^(n+1) - 20 + Lucas(2*n + 7))/5. - _Ehren Metcalfe_, Aug 21 2017
%F A214729 From _Colin Barker_, Aug 25 2017: (Start)
%F A214729 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) ).
%F A214729 a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4) for n>3. (End)
%F A214729 a(n) = A001654(n+3) - 2*(2 + (-1)^n). - _G. C. Greubel_, Dec 31 2023
%t A214729 With[{m = 6}, Table[Sum[Fibonacci[k + m]*Fibonacci[k], {k, 0, n}], {n, 0, 25}]] (* or *)
%t A214729 Table[(9 (-1)^(n + 1) - 20 + LucasL[2 n + 7])/5, {n, 0, 25}] (* _Michael De Vlieger_, Aug 23 2017 *)
%t A214729 LinearRecurrence[{3,0,-3,1},{0,13,34,102},40] (* _Harvey P. Dale_, Jun 13 2022 *)
%o A214729 (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
%o A214729 (Magma) [(9*(-1)^(n+1)-20+Lucas(2*n+7))/5: n in [0..40]]; // _Vincenzo Librandi_, Aug 26 2017
%o A214729 (SageMath) [fibonacci(n+3)*fibonacci(n+4) - 2*(2+(-1)^n) for n in range(41)] # _G. C. Greubel_, Dec 31 2023
%Y A214729 Cf. A001654, A027941, A059840(n+2), A064831, A080097, A080143 and A080144 for the m=0,1,...,5 members.
%Y A214729 Cf. A027941.
%K A214729 nonn,easy
%O A214729 0,2
%A A214729 _Wolfdieter Lang_, Jul 27 2012