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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

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

Index entries for sequences related to linear recurrences with constant coefficients

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, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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.

Adjacent sequences: A000075 A000076 A000077 this_sequence A000079 A000080 A000081

Sequence in context: A001383 A108564 A066369 this_sequence A034338 A166861 A026023

nonn,easy,nice

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

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