A001636 - OEIS

%I M0763 N0290 #44 Sep 08 2022 08:44:29

%S 0,2,3,6,10,17,21,38,57,92,143,225,351,555,868,1366,2142,3365,5282,

%T 8296,13023,20451,32108,50417,79160,124295,195159,306431,481139,

%U 755462,1186184,1862486,2924375,4591702,7209646,11320209,17774393,27908418,43820325

%N A Fielder sequence: a(n) = a(n-1) + a(n-2) - a(n-7), n >= 8.

%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="/A001636/b001636.txt">Table of n, a(n) for n = 1..1000</a>

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

%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_07">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 0, 0, 0, 0, -1).

%F G.f.: x^2*(2+x+x^2+x^3+x^4-6*x^5)/(1-x-x^2+x^7).

%F a(n) = a(n-2) + a(n-3) + a(n-4) + a(n-5) + a(n-6), n >= 7.

%p A001636:=-z*(2+3*z+4*z**2+5*z**3+6*z**4)/(z+1)/(z**5+z**3+z-1); # _Simon Plouffe_ in his 1992 dissertation

%p a:= n -> (Matrix([[6,-1$4,4,5]]). Matrix(7, (i,j)-> if (i=j-1) then 1 elif j=1 then [1$2,0$4,-1][i] else 0 fi)^n)[1,1]: seq(a(n), n=1..38); # _Alois P. Heinz_, Aug 01 2008

%t LinearRecurrence[{1, 1, 0, 0, 0, 0, -1}, {0, 2, 3, 6, 10, 17, 21}, 50] (* _T. D. Noe_, Aug 09 2012 *)

%o (PARI) a(n)=if(n<0,0,polcoeff(x^2*(2+x+x^2+x^3+x^4-6*x^5)/(1-x-x^2+x^7)+x*O(x^n),n))

%o (Magma) I:=[0, 2, 3, 6, 10, 17, 21]; [n le 7 select I[n] else Self(n-1) + Self(n-2) - Self(n-7): n in [1..30]]; // _G. C. Greubel_, Jan 09 2018

%Y Cf. A013983.

%K nonn

%O 1,2

%A _N. J. A. Sloane_

%E Edited by _Michael Somos_, Feb 17 2002