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4, 5, 8, 13, 20, 29, 40, 53, 68, 85, 104, 125, 148, 173, 200, 229, 260, 293, 328, 365, 404, 445, 488, 533, 580, 629, 680, 733, 788, 845, 904, 965, 1028, 1093, 1160, 1229, 1300, 1373, 1448, 1525, 1604, 1685, 1768, 1853, 1940, 2029, 2120, 2213, 2308, 2405, 2504
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| 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 recurrences 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)
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]
The only two real solutions of the form f(x)= A*x^p with positive p which satisfy f^(n)(x) = f^[ -1](x), x>=0, n>=1, with f^(n) the n-th derivative and f^[ -1] the compositional inverse of f, is obtained for p=p1(n)=(n+sqrt(a(n)))/2 and p=p2(n)=(n-sqrt(a(n)))/2, n>=1, and A=A(n)=(fallfac(p,n))^(-p/(p+1)),for p=p1(n) and p=p2(n), respectively. Here fallfac(x,k):=product(x-j,j=0..k-1), the falling factorials. See the T. Koshy reference, pp. 263-4 (there is also a solutions for negative p if n is even; see the corresponding comment in A002522). [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 21 2010]
(n + sqrt(a(n)))/2 = [n;n,n,...], with the regular continued fraction with period length 1. For a simple proof see. e.g., the Schroeder reference, pp 330-1. See also the first comment above.
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REFERENCES
| 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]
Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, New York, 2001. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 21 2010]
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LINKS
| Eric Weisstein's World of Mathematics, Near-Square Prime
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
| 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)=a(n-1)+2*n-1 (with a(0)=4) [From Vincenzo Librandi, Nov 22 2010]
G.f.: (4-7*x+5*x^2)/(1-3*x+3*x^2-x^3). [Colin Barker, Jan 06 2012]
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EXAMPLE
| 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
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MATHEMATICA
| a[n_] := n^2 + 4; (* From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 15 2008 *)
Range[0, 50]^2 + 4 (* From Harvey P. Dale, Jan 05 2011 *)
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PROG
| (Other) sage: [lucas_number1(3, n, -4) for n in xrange(0, 51)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 16 2009]
(PARI) a(n)=n^2+4 \\ Charles R Greathouse IV, Jun 10 2011
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CROSSREFS
| Cf. A005563, A046092, A001082, A002378, A036666, A062717, A028347.
Cf. A155965, A155966 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 08 2009]
Sequence in context: A174398 A030978 A101948 * A019526 A145488 A050892
Adjacent sequences: A087472 A087473 A087474 * A087476 A087477 A087478
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KEYWORD
| nonn,easy
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoogroups.com), Sep 09 2003
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EXTENSIONS
| More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Sep 14 2003
In the last comment line an end bracket was missing in my recent comment:A=A(n)=(fallfac(p,n))^(-p/(p+1),... I corrected it. Sorry. WL Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 28 2010
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