0,6

a(n)=number of compositions of n-3 with no part greater than 4. Example: a(7)=8 because we have 1+1+1+1=2+1+1=1+2+1=3+1=1+1+2=2+2=1+3=4. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 10 2004

a(n+4)=number of 0-1 sequences of length n that avoid 1111. - David Callan (callan(AT)stat.wisc.edu), Jul 19 2004

a(n)=number of matchings in the graph obtained by a zig-zag triangulation of a convex (n-3)-gon. Example: a(8)=15 because in the triangulation of the convex pentagon ABCDEA with diagonals AD and AC we have 15 matchings: the empty set, seven singletons and {AB,CD},{AB,DE},{BC,AD},{BC,DE},{BC,EA},{CD,EA} and {DE,AC}. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 25 2004

Number of permutations satisfying -k<=p(i)-i<=r, i=1..n-3, with k=1, r=3. - Vladimir Baltic (baltic(AT)matf.bg.ac.yu), Jan 17 2005

E. Deutsch, Problem 1613, Math. Mag., 75, No. 1, 64-64.

M. Feinberg, Fibonacci-Tribonacci, Fib. Quart. 1(#3) (1963), 71-74.

W. C. Lynch, The t-Fibonacci numbers and polyphase sorting, Fib. Quart., 8 (1970), pp. 6ff.

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).

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.4.

Problem 2803, Amer. Math. Monthly, 33 (1926), 229-232.

J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, Princeton, NJ, 1978.

T. D. Noe, Table of n, a(n) for n = 0..200

Joerg Arndt, Fxtbook

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 11

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.

Eric Weisstein's World of Mathematics, Tetranacci Number.

Index entries for sequences related to linear recurrences with constant coefficients

a(n) =A001630(n)-a(n-1) - Henry Bottomley

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

a(n) = term (1,4) in the 4x4 matrix [1,1,0,0; 1,0,1,0; 1,0,0,1; 1,0,0,0]^n. - Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jun 12 2008

G.f.: 1/(1-z-z^2-z^3-z^4). (S. Plouffe) [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 17 2009]

sage: taylor( mul(x/(1-x-x^2-x^3-x^4) for i in xrange(1,2)),x,0,33)#solution>> x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 15*x^6 + 29*x^7 +....+ 201061985*x^31 + 387559437*x^32 + 747044834*x^33+etc... [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 02 2009]

A000078:=-1/(-1+z+z**2+z**3+z**4); [S. Plouffe in his 1992 dissertation.]

a := n -> (Matrix([[1, 1, 0, 0], [1, 0, 1, 0], [1, 0, 0, 1], [1, 0, 0, 0]])^n)[1, 4]; seq ((a(n)), n=0..50); - Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jun 12 2008

g:=1/(1-z-z^2-z^3-z^4): gser:=series(g, z=0, 49): seq((coeff(gser, z, n)), n=-3..32); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 17 2009]

CoefficientList[Series[x^3/(1 - x - x^2 - x^3 - x^4), {x, 0, 50}], x]

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

Contribution from Cino Hilliard (hillcino368(AT)hotmail.com), Apr 29 2009: (Start)

(PARI) /* Simple Generation with 5 variables*/

g(n) =

{

local(a1=0, a2=0, a3=0, a4=1);

print1(a1", "a2", "a3", "a4", ");

for(x=5, n, a5=a1+a2+a3+a4;

print1(a5", ");

a1=a2; a2=a3; a3=a4; a4=a5;

)

}

(End)

(Other) sage: taylor( mul(x/(1-x-x^2-x^3-x^4) for i in xrange(1, 2)), x, 0, 33)# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 02 2009]

Row 4 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).

First differences are in A001631.

nonn,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Henry Bottomley (se16(AT)btinternet.com), Oct 09 2000

Definition augmented (with 4 initial terms) by Daniel Forgues (squid(AT)zensearch.com), Dec 02 2009