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A065885 a(n)-1, a(n) and a(n)+1 form three consecutive integers that can be factored into Fibonacci numbers. 3
2, 3, 4, 5, 9, 25, 26, 64, 169, 441, 1156, 3025, 7921, 20736, 54289, 142129, 372100, 974169, 2550409, 6677056, 17480761, 45765225, 119814916, 313679521, 821223649, 2149991424, 5628750625, 14736260449, 38580030724, 101003831721, 264431464441, 692290561600, 1812440220361 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

In general it can be shown that F(n-1)F(n+1), F(n)^2, F(n-2)F(n+2) form three consecutive increasing integers when n is odd and F(n-2)F(n+2), F(n)^2, F(n-1)F(n+1) for three consecutive increasing integers when n is even. Thus the sequence is infinite. [Corrected by Charles R Greathouse IV, Jul 17 2012]

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

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

FORMULA

Except for n = 1, 2, 4 and 7, a(n) is the square of a Fibonacci number.

From Colin Barker, Sep 30 2016: (Start) (based on the signature given in the link}

a(n) = 2*a(n-1)+2*a(n-2)-a(n-3) for n>10.

G.f.: x*(2-x-6*x^2-7*x^3-6*x^4+x^5-37*x^6-29*x^7+14*x^8+x^9) / ((1+x)*(1-3*x+x^2)).

(End)

EXAMPLE

440 = 8*55, 441 = 21^2, 442 = 13*34, so 441 is a term of the sequence.

PROG

(PARI) a(n)=if(n>7, fibonacci(n-2)^2, [2, 3, 4, 5, 9, 25, 26][n]) \\ Charles R Greathouse IV, Jul 17 2012

(PARI) Vec(x*(2-x-6*x^2-7*x^3-6*x^4+x^5-37*x^6-29*x^7+14*x^8+x^9)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Sep 30 2016

CROSSREFS

Cf. A065108, A000045, A007598.

Sequence in context: A018896 A162374 A323289 * A274331 A177064 A092233

Adjacent sequences:  A065882 A065883 A065884 * A065886 A065887 A065888

KEYWORD

nonn,easy

AUTHOR

John W. Layman, Nov 28 2001

STATUS

approved

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Last modified April 20 09:59 EDT 2019. Contains 322309 sequences. (Running on oeis4.)