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a(n) = F(n)^2 - F(n-1)^2 or F(n+1) * F(n-2) where F(n) = A000045(n), the Fibonacci numbers.
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%I #114 Jul 25 2023 04:03:09

%S 1,0,3,5,16,39,105,272,715,1869,4896,12815,33553,87840,229971,602069,

%T 1576240,4126647,10803705,28284464,74049691,193864605,507544128,

%U 1328767775,3478759201,9107509824,23843770275,62423800997,163427632720,427859097159,1120149658761

%N a(n) = F(n)^2 - F(n-1)^2 or F(n+1) * F(n-2) where F(n) = A000045(n), the Fibonacci numbers.

%C A001519(n)^2 = A079472(n)^2 + a(n)^2 and (A001519(n), A079472(n), a(n)) is a Pythagorean triple.

%C INVERT transform is A052156. PSUM transform is A007598. SUMADJ transform is A088305. BINOMIAL transform is A039717. BINOMIAL transform with 0 prepended is A112091 with 0 prepended. BINOMIAL transform inverse is A084179(n+1).

%H Vincenzo Librandi, <a href="/A226205/b226205.txt">Table of n, a(n) for n = 1..1000</a>

%H John P. Bonomo and Montana Ferita, <a href="https://doi.org/10.1080/07468342.2023.2225397">A Small Fib</a>, College Math. J., 2023.

%H Nurettin Irmak, <a href="https://doi.org/10.3906/mat-1605-127">Product of arbitrary Fibonacci numbers with distance 1 to Fibonomial coefficient</a>, Turk J Math, (2017) 41: 825-828. See p. 828.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).

%F G.f.: x * (1 - x)^2 / ((1 + x) * (1 -3*x + x^2)).

%F a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).

%F a(n) = - A121646(n).

%F a(n) = -a(1-n) for all n in Z.

%F a(n) = A121801(n+1) / 2. - _Michael Somos_, Jun 17 2014

%F a(n) = a(n-1) + A000045(n-1)^2 - 2*(-1)^n, for n>1. - _Alexander Samokrutov_, Sep 07 2015

%F a(n) = F(n-1)*F(n) - (-1)^n. - _Bruno Berselli_, Oct 30 2015

%F a(n) = 2^(-1-n)*(-(-1)^n*2^(3+n)-(3-sqrt(5))^n*(1+sqrt(5))+(-1+sqrt(5))*(3+sqrt(5))^n)/5. - _Colin Barker_, Sep 28 2016

%F From _Amiram Eldar_, Oct 06 2020: (Start)

%F Sum_{n>=3} 1/a(n) = (1/2) * A290565 - 1/4.

%F Sum_{n>=3} (-1)^(n+1)/a(n) = (3/2) * (1/phi - 1/2), where phi is the golden ratio (A001622). (End)

%e G.f. = x + 3*x^3 + 5*x^4 + 16*x^5 + 39*x^6 + 105*x^7 + 272*x^8 + 715*x^9 + ...

%p a:= n-> (<<0|1|0>, <0|0|1>, <-1|2|2>>^n. <<1,0,3>>)[1, 1]:

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Sep 28 2016

%t a[ n_] := Fibonacci[n + 1] Fibonacci[n - 2]; (* _Michael Somos_, Jun 17 2014 *)

%t CoefficientList[Series[(1 - x)^2/((1 + x) (1 - 3 x + x^2)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Jun 17 2014 *)

%o (Magma) [Fibonacci(n)^2-Fibonacci(n-1)^2: n in [1..40]]; // _Vincenzo Librandi_, Jun 18 2014

%o (PARI) {a(n) = fibonacci( n + 1) * fibonacci( n - 2)};

%o (PARI) a(n) = round(2^(-1-n)*(-(-1)^n*2^(3+n)-(3-sqrt(5))^n*(1+sqrt(5))+(-1+sqrt(5))*(3+sqrt(5))^n)/5) \\ _Colin Barker_, Sep 28 2016

%o (PARI) lista(nn) = {my(p = (3*x-1)/(x^3-2*x^2-2*x+1)); for (n=1, nn, p = deriv(p, x); print1(subst(p, x, 0)/n!, ", "); ); } \\ _Michel Marcus_, May 22 2018

%Y Cf. A001519, A001622, A001654, A007598, A033999, A039717, A052156, A079472, A084179, A088305, A112091, A121646, A290565.

%Y Cf. similar sequences of the type k*F(n)*F(n+1)+(-1)^n listed in A264080.

%Y Cf. A260259: numbers of the form F(n)*F(n+1)-(-1)^n. - _Bruno Berselli_, Nov 02 2015

%K nonn,easy

%O 1,3

%A _Michael Somos_, Jun 06 2013