%I A087475
%S A087475 4,5,8,13,20,29,40,53,68,85,104,125,148,173,200,229,260,293,328,365,404,
%T A087475 445,488,533,580,629,680,733,788,845,904,965,1028,1093,1160,1229,1300,
%U A087475 1373,1448,1525,1604,1685,1768,1853,1940,2029,2120,2213,2308,2405,2504
%N A087475 n^2 + 4.
%C A087475 Schroeder, p. 330, states "For positive n, these winding numbers are 
               precisely those whose continued fraction expansion is periodic and 
               has period length 1".
%C A087475 Sequence allows us to find X values of the equation: X^3 - 4*X^2 = Y^2. 
               To find Y values: b(n)=n*(n^2 + 4). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), 
               Nov 06 2007
%C A087475 Contribution from Artur Jasinski (grafix(AT)csl.pl), Oct 03 2008: (Start)
%C A087475 General formula for cotangent recurences type:
%C A087475 a(n+1)=a(n)^3+3*a(n) and a(1)=k is
%C A087475 a(n)=Floor[((k+Sqrt[k^2+4])/2)^(3^(n-1))] (*Artur Jasinski*) (End)
%C A087475 Except for the first term of [A087475] and [A155965], [A087475]^3^=[A155965]^2+[A155966]^2 
               [From Vncenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 08 2009]
%C A087475 a(n) = A156798(n)/A002522(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 16 2009]
%C A087475 Given sequences of the form S(n) = N*S(n-1) + S(n-2) starting (1, N,...), 
               having convergents with discriminant (N^2 + 4); S(p) == (a(n))^((p-1)/
               2)) mod p, for n>0, p = odd prime. Example: with N = 2 we have the 
               Pell series (1, 2, 5, 12, 29, 70, 169,..., with P(7) = 169. Then 
               169 == 8^(3) mod 7, with a(2) = 8. Cf. Schroeder, "Number Theory 
               in Science and Communication", p. 90, for N = 1: F(p) == 5^((p-1)/
               2)) mod p. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 23 2009]
%C A087475 a(n) = A156701(n)/A053755(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 13 2009]
%C A087475 Number of units of a(n) belongs to a periodic sequence: 4, 5, 8, 3, 0, 
               9, 0, 3, 8, 5.We conclude that a(n) and a(n+10) have the same number 
               of units. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 05 2009]
%D A087475 Manfred R. Schroeder, "Fractals, Chaos, Power Laws"; W.H. Freeman & Co, 
               1991, p. 330-331.
%D A087475 Manfred R. Schroeder, "Number Theory in Science and Communication", Springer 
               Verlag, 5-th ed., 2009. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Feb 23 2009]
%H A087475 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Near-SquarePrime.html">Near-Square Prime</a>
%F A087475 n^2 + 4 are discriminant terms in the formula for Positive Silver Mean 
               Constants, defined as barover[n], = [sqrt (n^2 + 4) - n]/2. Such 
               constants barover[n] = C, have the property: 1/C - C = n
%F A087475 a(n)=2*n+a(n-1)-3 (with a(1)=4) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 07 2009]
%e A087475 a(2) = 8, discriminant of algebraic representation of barover[2] = [2,
               2,2,...] = sqrt 2 - 1 = .41421356...= [(sqrt 8) - 2]/2. a(3) = 13, 
               discriminant of barover[3] = [3,3,3...] = .3027756... = [(sqrt 13) 
               - 3]/2
%e A087475 For n=2, a(2)=2*2+4-3=5; n=3, a(3)=2*3+5-3=8; n=4, a(4)=2*4+8-3=13 [From 
               Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 07 2009]
%t A087475 a[n_]:=n^2+4; [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 15 
               2008]
%o A087475 (Other) sage: [lucas_number1(3,n,-4) for n in xrange(0, 51)] # [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 16 2009]
%Y A087475 Cf. A005563, A046092, A001082, A002378, A036666, A062717, A028347.
%Y A087475 Cf. A155965, A155966 [From Vncenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Feb 08 2009]
%Y A087475 Sequence in context: A133940 A030978 A101948 this_sequence A019526 A145488 
               A050892
%Y A087475 Adjacent sequences: A087472 A087473 A087474 this_sequence A087476 A087477 
               A087478
%K A087475 nonn,new
%O A087475 0,1
%A A087475 Gary W. Adamson (qntmpkt(AT)yahoogroups.com), Sep 09 2003
%E A087475 More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Sep 14 
               2003