|
| |
|
|
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
|
Michael De Vlieger, Table of n, a(n) for n = 0..3036
Milica Anđelić and Carlos M. da Fonseca, On the constant coefficients of a certain recurrence relation: A simple proof, Heliyon (2021) Vol. 7, No. 8, e07764.
D. Panario, M. Sahin and 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; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 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: A089067 A339888 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
|
| |
|
|
