A001635 - OEIS

%I M0762 N0289 #53 Mar 19 2024 15:36:04

%S 0,2,3,6,10,11,21,30,48,72,110,171,260,401,613,942,1445,2216,3401,

%T 5216,8004,12278,18837,28899,44335,68018,104349,160089,245601,376791,

%U 578057,886830,1360538,2087279,3202216,4912704,7536863,11562737,17739062,27214520

%N A Fielder sequence: a(n) = a(n-1) + a(n-2) - a(n-6), n >= 7.

%C 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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H T. D. Noe, <a href="/A001635/b001635.txt">Table of n, a(n) for n = 1..1000</a>

%H Daniel C. Fielder, <a href="http://www.fq.math.ca/Scanned/6-3/fielder.pdf">Special integer sequences controlled by three parameters</a>, Fibonacci Quarterly 6, 1968, 64-70.

%H Daniel C. Fielder, <a href="http://www.fq.math.ca/Scanned/6-3/errata.pdf">Errata:Special integer sequences controlled by three parameters</a>, Fibonacci Quarterly 6, 1968, 64-70.

%H D. Fielder, <a href="/A027907/a027907_1.pdf">Letter to N. J. A. Sloane, Jun. 1991</a>

%H D. C. Fielder and C. O. Alford, <a href="/A001635/a001635.pdf">Simulation concepts for studying incomplete (but potentially recursive) sequences</a>, IASTED International Symposium Simulation and Modeling '89, Lugano, Switzerland, June 19-22, 1989. (Annotated scanned copy)

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,0,-1).

%F G.f.: x^2*(2 + x + x^2 + x^3 - 5*x^4)/(1 - x - x^2 + x^6).

%F a(n) = a(n-2) + a(n-3) + a(n-4) + a(n-5), n >= 6.

%F 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

%p 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

%p 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

%t LinearRecurrence[{1, 1, 0, 0, 0, -1}, {0, 2, 3, 6, 10, 11}, 50] (* _T. D. Noe_, Aug 09 2012 *)

%o (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))

%o (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

%Y Cf. A000129.

%K nonn

%O 1,2

%A _N. J. A. Sloane_

%E Edited by _Michael Somos_, Feb 17 2002