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 A089499 a(0)=0; a(1)=1; a(2n)=4*Sum_{k=0...n}a(2k-1); a(2n+1)=a(2n)+a(2n-1). 2
 0, 1, 4, 5, 24, 29, 140, 169, 816, 985, 4756, 5741, 27720, 33461, 161564, 195025, 941664, 1136689, 5488420, 6625109, 31988856, 38613965, 186444716, 225058681, 1086679440, 1311738121, 6333631924, 7645370045, 36915112104, 44560482149 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 1, 4, 5, 24, 29, 140,...= numerators in convergents to (sqrt(8) - 2) = continued fraction [0; 1, 4, 1, 4, 1, 4,...]; where sqrt(8) - 2 = .828427124... = the inradius of a right triangle with hypotenuse 6, legs sqrt(32) and 2. Denominators of convergents to [0; 1, 4, 1, 4, 1, 4,...] = A041011 starting (1, 5, 6, 29, 35,...). - Gary W. Adamson, Dec 22 2007 This is a strong divisibility sequence, that is, GCD(a(n),a(m)) = a(GCD(n,m)) for all natural numbers n and m. - Peter Bala, May 12 2014 LINKS Index entries for linear recurrences with constant coefficients, signature (0, 6, 0, -1). FORMULA For n>0, a(n)=A001333(n)+A084068(n-1)*(-1)^n; e.g. 29=41-12. a(n)*a(n+1)=A046729(n); cf. A001333. a(2n+1)=A001653(n); a(2n)=A005319(n). a(1) = 1, a(2n) = 4*a(2n-1) + a(2n-2); a(2n-1) = a(2n-2) + a(2n-3). Given the 2 X 2 matrix X = [1, 4; 1, 5], [a(2n-1), a(2n)] = top row of X^n. The sequence starting (1, 4, 5, 24, 29,...) = numerators in continued fraction [0; 1, 4, 1, 4, 1, 4,...] = (sqrt(8) - 2) = .828427124... E.g. X^3 = [29, 140; 35, 169], where 29/35, 140/169 are convergents to (sqrt(8)-2). - Gary W. Adamson, Dec 22 2007 a(n)=A000129(n)*A000034(n+1). a(n)=6*a(n-2)-a(n-4). G.f.: -x*(-1-4*x+x^2)/((x^2-2*x-1)*(x^2+2*x-1)). - R. J. Mathar, Jul 08 2009 From Peter Bala, May 12 2014: (Start) a(2*n + 1) = A041011(2*n + 1); a(2*n) = 4*A041011(2*n). For n odd, a(n) = (alpha^n - beta^n)/(alpha - beta), and for n even, a(n) = 4*(alpha^n - beta^n)/(alpha^2 - beta^2), where alpha = 1 + sqrt(2) and beta = 1 - sqrt(2). a(n) = product {j = 1..floor(n/2)} ( 4 + 4*cos^2(j*Pi/n) ) for n >= 1. (End) MATHEMATICA Numerator[NestList[(4/(4+#))&, 0, 60]] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2010 *) PROG (PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, 0, 6, 0]^n*[0; 1; 4; 5])[1, 1] \\ Charles R Greathouse IV, Nov 13 2015 CROSSREFS Cf. A041011. Sequence in context: A039583 A042123 A041531 * A249060 A042601 A219515 Adjacent sequences:  A089496 A089497 A089498 * A089500 A089501 A089502 KEYWORD nonn,easy AUTHOR Charlie Marion, Nov 11 2003 EXTENSIONS Corrected by T. D. Noe, Nov 08 2006 Definition corrected by Jonathan Sondow, Jun 06 2014 STATUS approved

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Last modified May 13 02:36 EDT 2021. Contains 343836 sequences. (Running on oeis4.)