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).

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

Offset: 0

Author

Charlie Marion, Nov 11 2003

Keywords

• nonn
• easy

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 = 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

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

• J. L. Ramirez, F. Sirvent, A q-Analogue of the Bi-Periodic Fibonacci Sequence, J. Int. Seq. 19 (2016) # 16.4.6, t_n at a=4, b=1.
• Index to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Crossrefs

Cf. A041011.

Programs

Mathematica

Numerator[NestList[(4/(4+#))&,0,60]] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2010 *)

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

Formula

For n > 0, a(n) = A001333(n) + A084068(n-1)*(-1)^n.

a(n)*a(n+1) = A046729(n).

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) = 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

From R. J. Mathar, Jul 08 2009: (Start)

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)). (End)

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)

Extensions

Corrected by T. D. Noe, Nov 08 2006

Definition corrected by Jonathan Sondow, Jun 06 2014