A226205 a(n) = F(n)^2 - F(n-1)^2 or F(n+1) * F(n-2) where F(n) = A000045(n), the Fibonacci numbers.

1, 0, 3, 5, 16, 39, 105, 272, 715, 1869, 4896, 12815, 33553, 87840, 229971, 602069, 1576240, 4126647, 10803705, 28284464, 74049691, 193864605, 507544128, 1328767775, 3478759201, 9107509824, 23843770275, 62423800997, 163427632720, 427859097159, 1120149658761

Offset

1,3

Comments

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

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).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

John P. Bonomo and Montana Ferita, A Small Fib, College Math. J., 2023.

Nurettin Irmak, Product of arbitrary Fibonacci numbers with distance 1 to Fibonomial coefficient, Turk J Math, (2017) 41: 825-828. See p. 828.

Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Formula

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

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

a(n) = - A121646(n).

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

a(n) = A121801(n+1) / 2. - Michael Somos, Jun 17 2014

a(n) = a(n-1) + A000045(n-1)^2 - 2*(-1)^n, for n>1. - Alexander Samokrutov, Sep 07 2015

a(n) = F(n-1)*F(n) - (-1)^n. - Bruno Berselli, Oct 30 2015

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

From Amiram Eldar, Oct 06 2020: (Start)

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

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

Example

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

Maple

a:= n-> (<<0|1|0>, <0|0|1>, <-1|2|2>>^n. <<1, 0, 3>>)[1, 1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 28 2016

Mathematica

a[ n_] := Fibonacci[n + 1] Fibonacci[n - 2]; (* Michael Somos, Jun 17 2014 *)
CoefficientList[Series[(1 - x)^2/((1 + x) (1 - 3 x + x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 17 2014 *)

Prog

(Magma) [Fibonacci(n)^2-Fibonacci(n-1)^2: n in [1..40]]; // Vincenzo Librandi, Jun 18 2014
(PARI) {a(n) = fibonacci( n + 1) * fibonacci( n - 2)};
(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
(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

Crossrefs

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

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

Cf. A260259: numbers of the form F(n)*F(n+1)-(-1)^n. - Bruno Berselli, Nov 02 2015

Sequence in context: A106588 A123785 A121646 * A300533 A221783 A368409

Adjacent sequences: A226202 A226203 A226204 * A226206 A226207 A226208

Keyword

nonn,easy

Author

Michael Somos, Jun 06 2013

Status

approved