A001634 - OEIS

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.
(Formerly M0746 N0281)

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

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).

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

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 *)

PROG

(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))

(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

(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 */