A001639 - OEIS

%I M3353 N1349 #44 Sep 08 2022 08:44:29

%S 1,1,4,9,16,22,36,65,112,186,309,522,885,1492,2509,4225,7124,12010,

%T 20236,34094,57453,96823,163163,274946,463316,780755,1315687,2217112,

%U 3736129,6295887,10609441,17878369,30127497,50768954,85552651,144167958,242942778

%N A Fielder sequence. a(n) = a(n-1) + a(n-3) + a(n-4) + a(n-5), n >= 6.

%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="/A001639/b001639.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_05">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 1, 1, 1).

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

%p A001639:=-(1+3*z**2+4*z**3+5*z**4)/(-1+z+z**3+z**4+z**5); # conjectured by _Simon Plouffe_ in his 1992 dissertation

%t Drop[CoefficientList[Series[x*(1+3*x^2+4*x^3+5*x^4)/(1-x-x^3-x^4-x^5),{x,0,40}], x], 1] (* _Stefan Steinerberger_, Apr 10 2006 *)

%t LinearRecurrence[{1,0,1,1,1}, {1,1,4,9,16}, 30] (* _G. C. Greubel_, Jan 09 2018 *)

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

%o (Magma) I:=[1, 1, 4, 9, 16]; [n le 5 select I[n] else Self(n-1) + Self(n-3) + Self(n-4) + Self(n-5): n in [1..30]]; // _G. C. Greubel_, Jan 09 2018

%Y Cf. A000570.

%K nonn

%O 1,3

%A _N. J. A. Sloane_

%E Edited by _Michael Somos_, Feb 17 2002