A001634 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.

0, 2, 3, 6, 5, 11, 14, 22, 30, 47, 66, 99, 143, 212, 308, 454, 663, 974, 1425, 2091, 3062, 4490, 6578, 9643, 14130, 20711, 30351, 44484, 65192, 95546, 140027, 205222, 300765, 440795, 646014, 946782, 1387574, 2033591, 2980370, 4367947, 6401535, 9381908

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

References

• E.-B. Escott, Reply to Query 1484, L'Intermédiaire des Mathématiciens, 8 (1901), 63-64.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n=0..500
• Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.
• Gregory T. Minton, Linear recurrence sequences satisfying congruence conditions, Proc. Amer. Math. Soc. 142 (2014), no. 7, 2337--2352. MR3195758.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 1).

Crossrefs

Cf. A013979, A107458.

Programs

Haskell

a001634 n = a001634_list !! n
a001634_list = 0 : 2 : 3 : 6 : zipWith (+) a001634_list
   (zipWith (+) (tail a001634_list) (drop 2 a001634_list))
-- Reinhard Zumkeller, Mar 23 2012

Maple

A001634:=-z*(2+3*z+4*z**2)/(1+z)/(z**3+z-1); # Simon Plouffe in his 1992 dissertation
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

Mathematica

LinearRecurrence[{0, 1, 1, 1}, {0, 2, 3, 6}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *)
CoefficientList[Series[x (2+3x+4x^2)/(1-x^2-x^3-x^4),{x,0,50}],x] (* Harvey P. Dale, Mar 26 2023 *)

Maxima

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 */

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))

Formula

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

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