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A001649 A Fielder sequence.
(Formerly M2649 N1056)
1

%I M2649 N1056

%S 1,3,7,15,26,57,106,207,403,788,1530,2985,5812,11322,22052,42959,

%T 83675,162993,317491,618440,1204651,2346534,4570791,8903409,17342876,

%U 33782050,65803777,128178646,249678140,486346022,947349461,1845334319,3594511719,7001720167

%N A Fielder sequence.

%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="/A001649/b001649.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 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.

%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, 1, 1, 0, 1).

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

%p A001649:=-(1+2*z+3*z**2+4*z**3+6*z**5)/(z+1)/(z**5-z**4+2*z**3-z**2+2*z-1); # [Conjectured by _Simon Plouffe_ in his 1992 dissertation.]

%t LinearRecurrence[{1, 1, 1, 1, 0, 1}, {1, 3, 7, 15, 26, 57}, 50] (* _T. D. Noe_, Aug 09 2012 *)

%t CoefficientList[Series[x*(1+2*x+3*x^2+4*x^3+6*x^5)/(1-x-x^2-x^3-x^4-x^6), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 19 2017 *)

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

%o (MAGMA) I:=[1,3,7,15,26,57]; [n le 6 select I[n] else Self(n-1) + Self(n-2) + Self(n-3) + Self(n-4) + Self(n-6): n in [1..30]]; // _G. C. Greubel_, Dec 19 2017

%K nonn

%O 1,2

%A _N. J. A. Sloane_

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Last modified May 6 09:43 EDT 2021. Contains 343580 sequences. (Running on oeis4.)