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A206351 a(n) = 7*a(n-1) - a(n-2) - 4 with a(1)=1, a(2)=3. 6
1, 3, 16, 105, 715, 4896, 33553, 229971, 1576240, 10803705, 74049691, 507544128, 3478759201, 23843770275, 163427632720, 1120149658761, 7677619978603, 52623190191456, 360684711361585, 2472169789339635 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A Pell sequence related to Heronian triangles (rational triangles), see A206334. The connection is this: consider the problem of finding triangles with area a positive integer n, and with sides (a, b, n) where a, b are rational. Note that n is both the area and one side. For many values of n this is not possible, and the sequence of such numbers n is quite erratic (see A206334). Nonetheless, each term in this sequence is such a value of n. For example, for n = 105 you can take the other two sides, a and b, to be 10817/104, and 233/104 and the area will equal n, i.e., 105.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

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

FORMULA

a(n) = 4/5+1/10*((7/2-3/2*sqrt(5))^(n-1)+(7/2+3/2*sqrt(5))^(n-1))+1/10*sqrt(5)*((7/2 +3/2*sqrt(5))^(n-1)-(7/2-3/2*sqrt(5))^(n-1)). - Paolo P. Lava, Feb 07 2012

From Bruno Berselli, Feb 07 2012: (Start)

G.f.: x*(1-5*x)/(1-8*x+8*x^2-x^3).

a(n) = A081018(n-1) + 1. (End)

a(n) = -A003482(-n) = Fibonacci(2*n)*Fibonacci(2*n-3). - Michael Somos, Jun 26 2018

a(n) = A089508(n-1) + 2 for n>1. - Bruno Berselli, Jun 20 2019 [Formula found by Umberto Cerruti]

EXAMPLE

G.f. = x + 3*x^2 + 16*x^3 + 105*x^4 + 715*x^5 + 4896*x^6 + 33553*x^7 + ... - Michael Somos, Jun 26 2018

MAPLE

genZ := proc(n)

local start;

option remember;

    start := [1, 3];

    if n < 3 then start[n]

    else 7*genZ(n - 1) - genZ(n - 2) - 4

    end if

end proc:

seq(genZ(n), n=1..20);

MATHEMATICA

LinearRecurrence[{8, -8, 1}, {1, 3, 16}, 50] (* Charles R Greathouse IV, Feb 07 2012 *)

RecurrenceTable[{a[1] == 1, a[2] == 3, a[n] == 7 a[n - 1] - a[n - 2] - 4}, a, {n, 20}] (* Bruno Berselli, Feb 07, 2012 *)

a[ n_] := Fibonacci[2 n] Fibonacci[2 n - 3]; (* Michael Somos, Jun 26 2018 *)

PROG

(PARI) Vec((1-5*x)/(1-8*x+8*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Feb 07 2012

(PARI) {a(n) = fibonacci(2*n) * fibonacci(2*n - 3)}; /* Michael Somos, Jun 26 2018 */

(Haskell)

a206351 n = a206351_list !! (n-1)

a206351_list = 1 : 3 : map (subtract 4)

               (zipWith (-) (map (* 7) (tail a206351_list)) a206351_list)

-- Reinhard Zumkeller, Feb 08 2012

(MAGMA) [Fibonacci(2*n)*Fibonacci(2*n-3): n in [1..30]]; // G. C. Greubel, Aug 12 2018

CROSSREFS

Cf. A000045, A003482, A089508.

Subsequence of A206334.

Sequence in context: A074542 A105622 A110903 * A085614 A215931 A271777

Adjacent sequences:  A206348 A206349 A206350 * A206352 A206353 A206354

KEYWORD

nonn,easy

AUTHOR

James R. Buddenhagen, Feb 06 2012

STATUS

approved

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Last modified November 17 01:05 EST 2019. Contains 329209 sequences. (Running on oeis4.)