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A002310 a(n) = 5*a(n-1) - a(n-2), with a(0) = 1 and a(1) = 2. 5

%I #56 Dec 30 2023 10:56:16

%S 1,2,9,43,206,987,4729,22658,108561,520147,2492174,11940723,57211441,

%T 274116482,1313370969,6292738363,30150320846,144458865867,

%U 692144008489,3316261176578,15889161874401,76129548195427,364758579102734,1747663347318243

%N a(n) = 5*a(n-1) - a(n-2), with a(0) = 1 and a(1) = 2.

%C 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

%C Limit_{n->infinity} a(n+1)/a(n) = (5 + sqrt(21))/2 = A107905. - _Wolfdieter Lang_, Nov 17 2023

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

%H Reinhard Zumkeller, <a href="/A002310/b002310.txt">Table of n, a(n) for n = 0..1000</a>

%H Margherita Maria Ferrari and Norma Zagaglia Salvi, <a href="https://www.emis.de/journals/JIS/VOL20/Salvi/salvi3.html">Aperiodic Compositions and Classical Integer Sequences</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.8.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H MathPages, <a href="http://www.mathpages.com/home/kmath334.htm">N = (x^2 + y^2)/(1+xy) is a Square</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-1).

%F 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

%F G.f.: (1-3x)/(1-5x+x^2). - _Philippe Deléham_, Nov 16 2008

%F a(n) = S(n, 5) - 3*S(n-1, 5), for n >= 0, with the S-Chebyshev polynomial (see A049310) S(n, 5) = A004254(n+1). - _Wolfdieter Lang_, Nov 17 2023

%t LinearRecurrence[{5, -1}, {1, 2}, 25] (* _T. D. Noe_, Feb 22 2014 *)

%o (Haskell)

%o a002310 n = a002310_list !! n

%o a002310_list = 1 : 2 :

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

%o -- _Reinhard Zumkeller_, Oct 16 2011

%o Cf. A049310, A004254.

%Y Cf. A002310, A002320, A004254, A049310, A049685, A054477, A107905.

%K nonn,easy

%O 0,2

%A Joe Keane (jgk(AT)jgk.org)

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Last modified March 29 08:13 EDT 2024. Contains 371265 sequences. (Running on oeis4.)