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A094586 Central numbers of the triangle T of all positive differences of distinct Fibonacci numbers.

%I A094586 #20 Dec 31 2024 15:39:04
%S A094586 1,5,16,47,131,356,953,2529,6676,17567,46135,121016,317201,831053,
%T A094586 2176712,5700303,14926171,39081404,102323209,267896585,701380076,
%U A094586 1836265535,4807451951,12586147632,32951083681,86267253461,225850919488
%N A094586 Central numbers of the triangle T of all positive differences of distinct Fibonacci numbers.
%C A094586 As T is also the triangle of sums of consecutive distinct Fibonacci numbers, a(n) is such a sum, namely Sum_{j=n+1..2n} Fibonacci(j).
%H A094586 Muniru A Asiru, <a href="/A094586/b094586.txt">Table of n, a(n) for n = 1..2000</a>
%H A094586 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,-2,1).
%F A094586 a(n) = Fibonacci(2n+2) - Fibonacci(n+2) = A094585(2n-1, n).
%F A094586 G.f.: x*(1+x-x^2)/((1-x-x^2)*(1-3*x+x^2)). - _Colin Barker_, Sep 16 2012
%e A094586 a(4) = F(10)-F(6) = 55-8 = 47.
%t A094586 Table[Sum[Fibonacci[n+i], {i,n}], {n,30}] (* _Zerinvary Lajos_, Jul 12 2009 *)
%t A094586 With[{F=Fibonacci}, Table[F[2n+2]-F[n+2], {n,30}]] (* _G. C. Greubel_, Jul 14 2019 *)
%t A094586 LinearRecurrence[{4,-3,-2,1},{1,5,16,47},30] (* _Harvey P. Dale_, Dec 31 2024 *)
%o A094586 (GAP) List([1..30],n->Fibonacci(2*n+2)-Fibonacci(n+2)); # _Muniru A Asiru_, Apr 28 2019
%o A094586 (PARI) vector(30, n, f=fibonacci; f(2*n+2)-f(n+2)) \\ _G. C. Greubel_, Jul 14 2019
%o A094586 (Magma) F:=Fibonacci; [F(2*n+2)-F(n+2): n in [1..30]]; // _G. C. Greubel_, Jul 14 2019
%o A094586 (Sage) f=fibonacci; [f(2*n+2)-f(n+2) for n in (1..30)] # _G. C. Greubel_, Jul 14 2019
%Y A094586 Cf. A000045, A094585.
%K A094586 nonn,easy
%O A094586 1,2
%A A094586 _Clark Kimberling_, May 13 2004