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%I #34 Mar 08 2023 11:53:32
%S 0,1,4,5,24,29,140,169,816,985,4756,5741,27720,33461,161564,195025,
%T 941664,1136689,5488420,6625109,31988856,38613965,186444716,225058681,
%U 1086679440,1311738121,6333631924,7645370045,36915112104,44560482149
%N a(0)=0; a(1)=1; a(2n) = 4*Sum_{k=0..n} a(2k-1); a(2n+1) = a(2n) + a(2n-1).
%C 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 = 0.828427124... = 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
%C 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
%H J. L. Ramirez, F. Sirvent, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Ramirez/ramirez9.html">A q-Analogue of the Bi-Periodic Fibonacci Sequence</a>, J. Int. Seq. 19 (2016) # 16.4.6, t_n at a=4, b=1.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-1).
%F For n > 0, a(n) = A001333(n) + A084068(n-1)*(-1)^n.
%F a(n)*a(n+1) = A046729(n).
%F a(2n+1) = A001653(n); a(2n) = A005319(n).
%F 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) = 0.828427124... 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
%F From _R. J. Mathar_, Jul 08 2009: (Start)
%F a(n) = A000129(n)*A000034(n+1).
%F a(n) = 6*a(n-2) - a(n-4).
%F G.f.: -x*(-1-4*x+x^2)/((x^2-2*x-1)*(x^2+2*x-1)). (End)
%F From _Peter Bala_, May 12 2014: (Start)
%F a(2*n + 1) = A041011(2*n + 1); a(2*n) = 4*A041011(2*n).
%F 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).
%F a(n) = Product_{j = 1..floor(n/2)} ( 4 + 4*cos^2(j*Pi/n) ) for n >= 1. (End)
%t Numerator[NestList[(4/(4+#))&,0,60]] (* _Vladimir Joseph Stephan Orlovsky_, Apr 13 2010 *)
%o (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
%Y Cf. A041011.
%K nonn,easy
%O 0,3
%A _Charlie Marion_, Nov 11 2003
%E Corrected by _T. D. Noe_, Nov 08 2006
%E Definition corrected by _Jonathan Sondow_, Jun 06 2014
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