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A002310 a(n) = 5*a(n-1) - a(n-2). 5
1, 2, 9, 43, 206, 987, 4729, 22658, 108561, 520147, 2492174, 11940723, 57211441, 274116482, 1313370969, 6292738363, 30150320846, 144458865867, 692144008489, 3316261176578, 15889161874401, 76129548195427, 364758579102734, 1747663347318243 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Together with A002320 these are the two sequences satisfying ( a(n)^2+a(n-1)^2 )/(1 - a(n)a(n-1)) is an integer, in both cases this integer is -5. - Floor van Lamoen, Oct 26 2001

REFERENCES

From a posting to Netnews group sci.math by ksbrown(AT)seanet.com (K. S. Brown) on Aug 15 1996.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Tanya Khovanova, Recursive Sequences

MathPages, N = (x^2 + y^2)/(1+xy) is a Square

Index entries for linear recurrences with constant coefficients, signature (5,-1).

FORMULA

Sequences A002310, A002320 and A049685 have this in common: each one satisfies a(n+1) = (a(n)^2+5)/a(n-1). - Graeme McRae, Jan 30 2005

G.f.: (1-3x)/(1-5x+x^2). [Philippe Deléham, Nov 16 2008]

a(n)=(1/42)*sqrt(21)*[(5/2)-(1/2)*sqrt(21)]^n-1/42*(5/2+1/2*sqrt(21))^n*sqrt(21)+(1/2)*[(5/2)+(1 /2)*sqrt(21)]^n+(1/2)*[(5/2)-(1/2)*sqrt(21)]^n, with n>=0. [Paolo P. Lava, Nov 21 2008]

MATHEMATICA

LinearRecurrence[{5, -1}, {1, 2}, 25] (* T. D. Noe, Feb 22 2014 *)

PROG

(Haskel)

a002310 n = a002310_list !! n

a002310_list = 1 : 2 :

   (zipWith (-) (map (* 5) (tail a002310_list)) a002310_list)

-- Reinhard Zumkeller, Oct 16 2011

CROSSREFS

Cf. A054477.

Sequence in context: A121365 A018960 A217666 * A055728 A006795 A055824

Adjacent sequences:  A002307 A002308 A002309 * A002311 A002312 A002313

KEYWORD

nonn,easy

AUTHOR

Joe Keane (jgk(AT)jgk.org)

STATUS

approved

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Last modified August 23 22:23 EDT 2017. Contains 291021 sequences.