OFFSET
1,2
COMMENTS
This is an application of the general formula that Paul Barry gives for sequence A000129 to the subsequence of odd-indexed terms. - Pat Costello (pat.costello(AT)eku.edu), May 20 2003
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.
Daniel C. Fielder, Errata:Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.
D. Fielder, Letter to N. J. A. Sloane, Jun. 1991
D. C. Fielder and C. O. Alford, Simulation concepts for studying incomplete (but potentially recursive) sequences, IASTED International Symposium Simulation and Modeling '89, Lugano, Switzerland, June 19-22, 1989. (Annotated scanned copy)
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
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,0,-1).
FORMULA
G.f.: x^2*(2 + x + x^2 + x^3 - 5*x^4)/(1 - x - x^2 + x^6).
a(n) = a(n-2) + a(n-3) + a(n-4) + a(n-5), n >= 6.
a(n) = Sum_{k=0..n} C(2*n+1, 2*k+1) * 2^k. - Pat Costello (pat.costello(AT)eku.edu), May 20 2003
MAPLE
A001635:=-z*(2+3*z+4*z**2+5*z**3)/(-1+z**2+z**3+z**4+z**5); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation
a := n -> (Matrix([[5, -1$3, 3, 4]]). Matrix(6, (i, j)-> if (i=j-1) then 1 elif j=1 then [1$2, 0$3, -1][i] else 0 fi)^n)[1, 1] ; seq (a(n), n=1..39); # Alois P. Heinz, Aug 01 2008
MATHEMATICA
LinearRecurrence[{1, 1, 0, 0, 0, -1}, {0, 2, 3, 6, 10, 11}, 50] (* T. D. Noe, Aug 09 2012 *)
PROG
(PARI) a(n)=if(n<0, 0, polcoeff(x^2*(2+x+x^2+x^3-5*x^4)/(1-x-x^2+x^6)+x*O(x^n), n))
(Magma) I:=[0, 2, 3, 6, 10, 11]; [n le 6 select I[n] else Self(n-1) + Self(n-2) - Self(n-6): n in [1..30]]; // G. C. Greubel, Jan 09 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by Michael Somos, Feb 17 2002
STATUS
approved