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A007805 a(n) = Fibonacci(6n+3)/2. 33
1, 17, 305, 5473, 98209, 1762289, 31622993, 567451585, 10182505537, 182717648081, 3278735159921, 58834515230497, 1055742538989025, 18944531186571953, 339945818819306129, 6100080207560938369 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Hypotenuse (z) of Pythagorean triples (x,y,z) with |2x-y|=1.

x(n) := 2*A049629(n) and y(n) := a(n), n>= 0,  give all positive solutions of the Pell equation x^2 - 5*y^2 = -1. See the Gregory V. Richardson formula, where his x is the y here and A075796(n+1) = x(n). - Wolfdieter Lang, Jun 20 2013

Positive numbers n such that 5*n^2 - 1 is a square (A075796(n+1)^2). - Gregory V. Richardson, Oct 13 2002

LINKS

T. D. Noe, Table of n, a(n) for n=0..100

Tanya Khovanova, Recursive Sequences

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume

Index entries for linear recurrences with constant coefficients, signature (18,-1).

Index entries for sequences related to Chebyshev polynomials.

FORMULA

G.f.: (1-x)/(1-18*x+x^2). a(n)=18*a(n-1)-a(n-2), n>1. a(0)=1, a(1)=17.

a(n) = A001076(2n+1).

a(n+1) = 9*a(n)+4*(5*a(n)^2-1)^0.5. - Richard Choulet, Aug 30 2007, Dec 28 2007

a(n) = ((2+sqrt(5))^(2*n+1)-(2-sqrt(5))^(2*n+1))/(2*sqrt(5)). - Dean Hickerson, Dec 09 2002

a(n) ~ (1/10)*sqrt(5)*(sqrt(5) + 2)^(2*n+1). - Joe Keane (jgk(AT)jgk.org), May 15 2002

Lim. n->Inf. a(n)/a(n-1) = 8*phi + 5 = 9 + 4*Sqrt(5). - Gregory V. Richardson, Oct 13 2002

Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then a(n)=q(n, 16). - Benoit Cloitre, Dec 06 2002

a(n) = 19*a(n-1)- 19*a(n-2) + a(n-3); f(x) = (sqrt(5)/10)*((2+sqrt(5))*(9+4*sqrt(5))^(x-1) - (2-sqrt(5))*(9-4*sqrt(5))^(x-1)). - Antonio Alberto Olivares, May 15 2008

a(n) = 17a(n-1) + 17a(n-2) - a(n-3). - Antonio Alberto Olivares, Jun 19 2008

a(n) = b(n+1) - b(n), n >=0, with b(n) := F(6*n)/F(6) = A049660(n). First differences. See the o.g.f.s. - Wolfdieter Lang, 2012

a(n) = S(n,18)- S(n-1,18) with the Chebyshev S-polynomials (A049310). - Wolfdieter Lang, Jun 20 2013

sum {n >= 1} 1/( a(n) - 1/a(n) ) = 1/4^2. Compare with A001519 and A097843. - Peter Bala, Nov 29 2013

a(n) = 9*a(n-1) + 8*A049629(n-1), n>=1, a(0) = 1. This is just the rewritten Chebyshev S(n, 18) recurrence. - Wolfdieter Lang, Aug 26 2014

From Peter Bala, Mar 23 2015: (Start)

a(n) = ( Fibonacci(6*n + 6 - 2*k) - Fibonacci(6*n + 2*k) )/( Fibonacci(6 - 2*k) - Fibonacci(2*k) ), for k an arbitrary integer.

a(n) = ( Fibonacci(6*n + 6 - 2*k - 1) + Fibonacci(6*n + 2*k + 1) )/( Fibonacci(6 - 2*k - 1) + Fibonacci(2*k + 1) ), for k an arbitrary integer.

The aerated sequence (b(n))n>=1 = [1, 0, 17, 0, 305, 0, 5473, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -20, Q = 1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials. (End)

a(n) = sqrt(2+(9-4*sqrt(5))^(1+2*n)+(9+4*sqrt(5))^(1+2*n))/(2*sqrt(5)). - Gerry Martens, Jun 04 2015

MAPLE

seq(combinat:-fibonacci(6*n+3)/2, n=0..30); # Robert Israel, Sep 10 2014

MATHEMATICA

LinearRecurrence[{18, -1}, {1, 17}, 50] (* Sture Sjöstedt, Nov 29 2011 *)

Table[Fibonacci[6n+3]/2, {n, 0, 20}] (* Harvey P. Dale, Dec 17 2011 *)

PROG

(Haskell)

a007805 = (`div` 2) . a000045 . (* 3) . (+ 1) . (* 2)

-- Reinhard Zumkeller, Mar 26 2013

(PARI) a(n)=fibonacci(6*n+3)/2 \\ Edward Jiang, Sep 09 2014

CROSSREFS

Cf. A000045.

Row 18 of array A094954.

Row 2 of array A188647.

Cf. similar sequences listed in A238379.

Sequence in context: A163049 A083453 A090437 * A158585 A201232 A156085

Adjacent sequences:  A007802 A007803 A007804 * A007806 A007807 A007808

KEYWORD

nonn,nice,easy

AUTHOR

James A. Raymond, Clark Kimberling

EXTENSIONS

Better description and more terms from Michael Somos.

STATUS

approved

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Last modified May 28 00:50 EDT 2017. Contains 287210 sequences.