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A005824 a(n) = 5a(n-2) - 2a(n-4).
(Formerly M2489)
5
0, 1, 1, 3, 5, 13, 23, 59, 105, 269, 479, 1227, 2185, 5597, 9967, 25531, 45465, 116461, 207391, 531243, 946025, 2423293, 4315343, 11053979, 19684665, 50423309, 89792639, 230008587, 409593865, 1049196317, 1868384047, 4785964411 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

REFERENCES

D. Panario, M. Sahin, Q. Wang, A family of Fibonacci-like conditional sequences, INTEGERS, Vol. 13, 2013, #A78.

Shallit, Jeffrey; On the worst case of three algorithms for computing the Jacobi symbol. J. Symbolic Comput. 10 (1990), no. 6, 593-610.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=0..31.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for linear recurrences with constant coefficients, signature (0,5,0,-2).

FORMULA

Also a(n) = a(n-1) + 2a(n-2) if n is even, else a(n) = 2a(n-1) + a(n-2).

Comment from Paolo P. Lava, Jun 10 2008: (Start) a(n) = (1/68) * (-1)^n * [5/2 + (1/2) * sqrt(17)]^(-1/4) * [5/2 + (1/2) * sqrt(17)]^[(1/4) * (-1)^n] * [5/2 + (1/2) * sqrt(17)]^[(1/2) * n] * sqrt(17) - (1/68) * [5/2 - (1/2) * sqrt(17)]^(-1/4) * (-1)^n * sqrt(17) * [5/2 - (1/2) * sqrt(17)]^[(1/4) * (-1)^n] * [5/2 - (1/2) * sqrt(17)]^[(1/2) * n] +

(1/4) * [5/2 - (1/2) * sqrt(17)]^( - 1/4) * [5/2 - (1/2) * sqrt(17)]^[(1/4) * (-1)^n] * [5/2 - (1/2) * sqrt(17)]^[(1/2) * n] + (1/4) * [5/2 + (1/2) * sqrt(17)]^(-1/4) * [5/2 + (1/2) * sqrt(17)]^[(1/4) * (-1)^n] * [5/2 + (1/2) * sqrt(17)]^[(1/2) * n] - (1/4) * (-1)^n * [5/2 + (1/2) * sqrt(17)]^(-1/4) * [5/2 + (1/2) * sqrt(17)]^[(1/4) * (-1)^n] * [5/2 + (1/2) * sqrt(17)]^[(1/2) * n] - (1/4) * [5/2 - (1/2) * sqrt(17)]^(-1/4) * (-1)^n * [5/2 - (1/2) * sqrt(17)]^[(1/4) * (-1)^n] * [5/2 - (1/2) * sqrt(17)]^[(1/2) * n] +

(3/68) * [5/2 + (1/2) * sqrt(17)]^(-1/4) * [5/2 + (1/2) * sqrt(17)]^[(1/4) * (-1)^n] * [5/2 + (1/2) * sqrt(17)]^[(1/2) * n] * sqrt(17) - (3/68) * [5/2 - (1/2) * sqrt(17)]^(-1/4) * sqrt(17) * [5/2 - (1/2) * sqrt(17)]^[(1/4) * (-1)^n] * [5/2 - (1/2) * sqrt(17)]^[(1 /2) * n], with n>= 0 (End)

a(2n+1)=A052984(n). a(2n)=A107839(n-1). [Index corrected by R. J. Mathar, Apr 01 2009]

MAPLE

A005824:=-z*(2*z+1)*(z-1)/(1-5*z**2+2*z**4); [Simon Plouffe in his 1992 dissertation.]

MATHEMATICA

a[0] = 0; a[1] = 1; a[n_] := a[n] = If[ EvenQ[n], a[n - 1] + 2a[n - 2], 2a[n - 1] + a[n - 2]]; Table[a[n], {n, 0, 31}]

LinearRecurrence[{0, 5, 0, -2}, {0, 1, 1, 3}, 40] (* Harvey P. Dale, Jul 09 2015 *)

CROSSREFS

Cf. A079162.

Sequence in context: A045414 A089067 A026733 * A027305 A026766 A026709

Adjacent sequences:  A005821 A005822 A005823 * A005825 A005826 A005827

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jeffrey Shallit

EXTENSIONS

Extended by Robert G. Wilson v, Dec 29 2002

Corrected index in formula referring to A052984. Added formula with A107839 R. J. Mathar, Apr 01 2009

STATUS

approved

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Last modified March 26 10:42 EDT 2017. Contains 284111 sequences.