A001634 - OEIS

%I M0746 N0281 #58 Jan 05 2025 19:51:32

%S 0,2,3,6,5,11,14,22,30,47,66,99,143,212,308,454,663,974,1425,2091,

%T 3062,4490,6578,9643,14130,20711,30351,44484,65192,95546,140027,

%U 205222,300765,440795,646014,946782,1387574,2033591,2980370,4367947,6401535,9381908

%N a(n) = a(n-2) + a(n-3) + a(n-4), with initial values a(0) = 0, a(1) = 2, a(2) = 3, a(3) = 6.

%D E.-B. Escott, Reply to Query 1484, L'Intermédiaire des Mathématiciens, 8 (1901), 63-64.

%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="/A001634/b001634.txt">Table of n, a(n) for n=0..500</a>

%H Daniel C. Fielder, <a href="https://web.archive.org/web/2024*/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 Gregory T. Minton, <a href="http://dx.doi.org/10.1090/S0002-9939-2014-12168-X">Linear recurrence sequences satisfying congruence conditions</a>, Proc. Amer. Math. Soc. 142 (2014), no. 7, 2337--2352. MR3195758.

%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_04">Index entries for linear recurrences with constant coefficients</a>, signature (0, 1, 1, 1).

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

%F a(n) = Sum_{k=0..(n-1)/2}(Sum_{j=0..k+1}(binomial(j,n-2*k-j-1)*binomial(k+1,j))/(k+1))*(n+1). - _Vladimir Kruchinin_, Mar 22 2016

%p A001634:=-z*(2+3*z+4*z**2)/(1+z)/(z**3+z-1); # _Simon Plouffe_ in his 1992 dissertation

%p a:= n-> (Matrix([[0,4,-1,-1]]). Matrix(4, (i,j)-> if (i=j-1) then 1 elif j=1 then [0,1,1,1][i] else 0 fi)^n)[1,1]: seq(a(n), n=0..40); # _Alois P. Heinz_, Aug 01 2008

%t LinearRecurrence[{0, 1, 1, 1}, {0, 2, 3, 6}, 100] (* _Vladimir Joseph Stephan Orlovsky_, Jul 01 2011 *)

%t CoefficientList[Series[x (2+3x+4x^2)/(1-x^2-x^3-x^4),{x,0,50}],x] (* _Harvey P. Dale_, Mar 26 2023 *)

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

%o (Haskell)

%o a001634 n = a001634_list !! n

%o a001634_list = 0 : 2 : 3 : 6 : zipWith (+) a001634_list

%o    (zipWith (+) (tail a001634_list) (drop 2 a001634_list))

%o -- _Reinhard Zumkeller_, Mar 23 2012

%o (Maxima)

%o a(n):=(sum(sum(binomial(j,n-2*k-j-1)*binomial(k+1,j),j,0,k+1)/(k+1),k,0,(n-1)/2))*(n+1); /* _Vladimir Kruchinin_, Mar 22 2016 */

%Y Cf. A013979, A107458.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_