A005824 a(n) = 5*a(n-2) - 2*a(n-4), with initial terms 0,1,1,3.

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, 8522732505, 21831429421, 38876894431

Offset: 0

Author

N. J. A. Sloane, Jeffrey Shallit

Keywords

• nonn
• easy

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.
• Sean A. Irvine, Walks on Graphs.
• 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
• Jeffrey 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).

Crossrefs

Cf. A079162.

Programs

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 *)

Formula

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

g.f.: -x*(2*x+1)*(x-1)/(1-5*x^2+2*x^4). Simon Plouffe in his 1992 dissertation.

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]

a(n) = A109165(n-1)-A109165(n-2). - R. J. Mathar, Jan 13 2025

Extensions

Extended by Robert G. Wilson v, Dec 29 2002