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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

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.

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

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.

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

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).

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). [Index corrected by R. J. Mathar, Apr 01 2009]

a(2n) = A107839(n-1). [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 * A336103 A027305 A026766

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

STATUS

approved

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Last modified November 30 05:31 EST 2020. Contains 338781 sequences. (Running on oeis4.)