A003482 a(n) = 7*a(n-1) - a(n-2) + 4, with a(0) = 0, a(1) = 5. (Formerly M3988)

0, 5, 39, 272, 1869, 12815, 87840, 602069, 4126647, 28284464, 193864605, 1328767775, 9107509824, 62423800997, 427859097159, 2932589879120, 20100270056685, 137769300517679, 944284833567072, 6472224534451829, 44361286907595735, 304056783818718320

Offset

0,2

Comments

The values (a(n),x(n)), n >= 2, x(n)=Fibonacci(2*n+2)*Fibonacci(2*n+3)=A081018(n+1), are the integer solutions (a,x) of the equation binomial(x+1,a+1) + binomial(x+2,a+1) = binomial(x+3,a+1). - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

The values (a(n),x(n)), n >= 2 are also the integer solutions (a, x) of the equation x(a+1) = (x-a)(x-a-1) or, equivalently, binomial(x, a) = binomial(x-1, a+1). - Tomohiro Yamada, May 30 2018

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Heiko Harborth, Fermat-like binomial equations, Applications of Fibonacci numbers, Proc. 2nd Int. Conf., San Jose, California, August 1986, 1-5 (1988).

Sébastien Labbé and Jana Lepšová, A Fibonacci's complement numeration system, arXiv:2205.02574 [cs.FL], 2022.

Sébastien Labbé and Jana Lepšová, A Fibonacci analogue of the two's complement numeration system, RAIRO-Theor. Inf. Appl. (2023) Vol. 57, No. 12. See p. 16.

D. A. Lind, The quadratic field Q(sqrt(5)) and a certain diophantine equation, Fibonacci Quart. 6(3) (1968), 86-93.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

John Riordan and N. J. A. Sloane, Correspondence, 1974

David Singmaster, Repeated binomial coefficients and Fibonacci numbers, Fibonacci Quart. 13 (1973), 295-298.

S. M. Tanny and M. Zuker, On a unimodal sequence of binomial coefficients, Discrete Math. 9 (1974), 79-89.

Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Formula

a(n) = Fibonacci(2*n) * Fibonacci(2*n+3).

a(n) = Fibonacci(2*n+2)^2 - Fibonacci(2*n+1)^2. - Gary Detlefs, Oct 12 2011

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3). - Vladimir Joseph Stephan Orlovsky and Vincenzo Librandi, Jan 22 2012

a(n) = -4/5 + (sqrt(5)/5 + 2/5)*(7/2 + 3*sqrt(5)/2)^n - (sqrt(5)/5 - 2/5)*(7/2 - 3*sqrt(5)/2)^n. - Antonio Alberto Olivares, May 29 2013

a(n) = -A206351(-n) for all n in Z. - Michael Somos, Jun 26 2018

From Sébastien Labbé, May 06 2022: (Start)

a(n) = Sum_{k=2..2*n+1} Fibonacci(k)^2.

a(n) = A001654(2*n+1)-1. (End)

Example

G.f. = 5*x + 39*x^2 + 272*x^3 + 1869*x^4 + 12815*x^5 + 87840*x^6 + ... - Michael Somos, Jun 26 2018

Maple

A003482:=z*(-5+z)/(z-1)/(z**2-7*z+1); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

LinearRecurrence[{8, -8, 1}, {0, 5, 39}, 30] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)

Prog

(PARI) a(n)=fibonacci(2*n)*fibonacci(2*n+3) \\ Charles R Greathouse IV, May 29 2013

Crossrefs

Cf. A000045, A001109, A001654, A081018, A206351.

Sequence in context: A075135 A202391 A053573 * A221357 A201442 A135849

Adjacent sequences: A003479 A003480 A003481 * A003483 A003484 A003485

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved