0,1

Schroeder, p. 330, states "For positive n, these winding numbers are precisely those whose continued fraction expansion is periodic and has period length 1".

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

Contribution from Artur Jasinski (grafix(AT)csl.pl), Oct 03 2008: (Start)

General formula for cotangent recurences type:

a(n+1)=a(n)^3+3*a(n) and a(1)=k is

a(n)=Floor[((k+Sqrt[k^2+4])/2)^(3^(n-1))] (*Artur Jasinski*) (End)

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]

a(n) = A156798(n)/A002522(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 16 2009]

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]

a(n) = A156701(n)/A053755(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 13 2009]

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]

Manfred R. Schroeder, "Fractals, Chaos, Power Laws"; W.H. Freeman & Co, 1991, p. 330-331.

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]

Eric Weisstein's World of Mathematics, Near-Square Prime

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

a(n)=2*n+a(n-1)-3 (with a(1)=4) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 07 2009]

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

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]

a[n_]:=n^2+4; [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 15 2008]

(Other) sage: [lucas_number1(3, n, -4) for n in xrange(0, 51)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 16 2009]

Cf. A005563, A046092, A001082, A002378, A036666, A062717, A028347.

Cf. A155965, A155966 [From Vncenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 08 2009]

Sequence in context: A133940 A030978 A101948 this_sequence A019526 A145488 A050892

Adjacent sequences: A087472 A087473 A087474 this_sequence A087476 A087477 A087478

nonn,new

Gary W. Adamson (qntmpkt(AT)yahoogroups.com), Sep 09 2003

More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Sep 14 2003