This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A087265 #56 Apr 19 2025 06:14:02
%S A087265 2,47,2207,103682,4870847,228826127,10749957122,505019158607,
%T A087265 23725150497407,1114577054219522,52361396397820127,
%U A087265 2459871053643326447,115561578124838522882,5428934300813767249007,255044350560122222180447
%N A087265 Lucas numbers L(8*n).
%C A087265 a(n+1)/a(n) converges to (47+sqrt(2205))/2 = 46.9787137... a(0)/a(1)=2/47; a(1)/a(2)=47/2207; a(2)/a(3)=2207/103682; a(3)/a(4)=103682/4870847; etc. Lim_{n->infinity} a(n)/a(n+1) = 0.02128623625... = 2/(47+sqrt(2205)) = (47-sqrt(2205))/2.
%C A087265 a(n) = a(-n). - _Alois P. Heinz_, Aug 07 2008
%C A087265 From _Peter Bala_, Oct 14 2019: (Start)
%C A087265 Let F(x) = Product_{n >= 0} (1 + x^(4*n+1))/(1 + x^(4*n+3)). Let Phi = 1/2*(sqrt(5) - 1). This sequence gives the partial denominators in the simple continued fraction expansion of the number F(Phi^8) = 1.0212763906... = 1 + 1/(47 + 1/(2207 + 1/(103682 + ...))).
%C A087265 Also F(-Phi^8) = 0.9787231991... has the continued fraction representation 1 - 1/(47 - 1/(2207 - 1/(103682 - ...))) and the simple continued fraction expansion 1/(1 + 1/((47 - 2) + 1/(1 + 1/((2207 - 2) + 1/(1 + 1/((103682 - 2) + 1/(1 + ...))))))).
%C A087265 F(Phi^8)*F(-Phi^8) = 0.9995468962... has the simple continued fraction expansion 1/(1 + 1/((47^2 - 4) + 1/(1 + 1/((2207^2 - 4) + 1/(1 + 1/((103682^2 - 4) + 1/(1 + ...))))))).
%C A087265 1/2 + 1/2*F(Phi^8)/F(-Phi^8) = 1.0217391349... has the simple continued fraction expansion 1 + 1/((47 - 2) + 1/(1 + 1/((103682 - 2) + 1/(1 + 1/(228826127 - 2) + 1/(1 + ...))))). (End)
%D A087265 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 91.
%D A087265 R. P. Stanley. Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999.
%H A087265 Indranil Ghosh, <a href="/A087265/b087265.txt">Table of n, a(n) for n = 0..596</a>
%H A087265 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A087265 A. V. Zarelua, <a href="https://doi.org/10.1007/s11006-006-0090-y">On Matrix Analogs of Fermat's Little Theorem</a>, Mathematical Notes, vol. 79, no. 6, 2006, pp. 783-796. Translated from Matematicheskie Zametki, vol. 79, no. 6, 2006, pp. 840-855.
%H A087265 <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>
%H A087265 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (47,-1).
%F A087265 a(n) = 47*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 47.
%F A087265 a(n) = ((47+sqrt(2205))/2)^n + ((47-sqrt(2205))/2)^n
%F A087265 (a(n))^2 = a(2n)+2.
%F A087265 G.f.: (2-47*x)/(1-47*x+x^2). - _Alois P. Heinz_, Aug 07 2008
%F A087265 From _Peter Bala_, Oct 14 2019: (Start)
%F A087265 a(n) = F(8*n+8)/F(8) - F(8*n-8)/F(8) = A049668(n+1) - A049668(n-1).
%F A087265 a(n) = trace(M^n), where M is the 2 X 2 matrix [0, 1; 1, 1]^8 = [13, 21; 21, 34].
%F A087265 Consequently the Gauss congruences hold: a(n*p^k) = a(n*p^(k-1)) ( mod p^k ) for all prime p and positive integers n and k. See Zarelua and also Stanley (Ch. 5, Ex. 5.2(a) and its solution).
%F A087265 45*Sum_{n >= 1} 1/(a(n) - 49/a(n)) = 1: (49 = Lucas(8) + 2 and 45 = Lucas(8) - 2)
%F A087265 49*Sum_{n >= 1} (-1)^(n+1)/(a(n) + 45/a(n)) = 1.
%F A087265 x*exp(Sum_{n >= 1} a(n)*x^/n) = x + 47*x^2 + 2208*x^3 + ... is the o.g.f. for A049668. (End)
%F A087265 E.g.f.: 2*exp(47*x/2)*cosh(21*sqrt(5)*x/2). - _Stefano Spezia_, Oct 18 2019
%F A087265 From _Peter Bala_, Apr 16 2025: (Start)
%F A087265 a(n) = Lucas(2*n)^4 - 4*Lucas(2*n)^2 + 2 = 2*T(4, (1/2)*Lucas(2*n)), where T(k, x) denotes the k-th Chebyshev polynomial of the first kind; more generally, for k >= 0, Lucas(2*k*n) = 2*T(k, Lucas(2*n)/2).
%F A087265 Sum_{n >= 1} 1/a(n) = (1/4) * (theta_3( (47 - sqrt(2205))/2 )^2 - 1) and
%F A087265 Sum_{n >= 1} (-1)^(n+1)/a(n) = (1/4) * (1 - theta_3( (sqrt(2205) - 47)/2 )^2),
%F A087265 where theta_3(x) = 1 + 2*Sum_{n >= 1} x^(n^2) (see A000122). See Borwein and Borwein, Proposition 3.5 (i), p. 91. Cf. A153415 and A003499. (End)
%e A087265 a(4) = 4870847 = 47*a(3) - a(2) = 47*103682 - 2207=((47+sqrt(2205))/2)^4 + ( (47-sqrt(2205))/2)^4 =4870846.999999794696 + 0.000000205303 = 4870847.
%p A087265 a:= n-> (Matrix([[2,47]]). Matrix([[47,1],[ -1,0]])^(n))[1,1]:
%p A087265 seq(a(n), n=0..14); # _Alois P. Heinz_, Aug 07 2008
%t A087265 LucasL[8*Range[0,20]] (* or *) LinearRecurrence[{47,-1},{2,47},20] (* _Harvey P. Dale_, Oct 23 2017 *)
%o A087265 (Magma) [ Lucas(8*n) : n in [0..100]]; // _Vincenzo Librandi_, Apr 14 2011
%Y A087265 Cf. A000032. Cf. Lucas(k*n): A005248 (k = 2), A014448 (k = 3), A056854 (k = 4), A001946 (k = 5), A087215 (k = 6), A087281 (k = 7), A087287 (k = 9), A065705 (k = 10), A089772 (k = 11), A089775 (k = 12).
%Y A087265 a(n) = A000032(8n).
%K A087265 easy,nonn
%O A087265 0,1
%A A087265 Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 19 2003
%E A087265 Terms a(22)-a(27) from _John W. Layman_, Jun 14 2004