A017817 a(n) = a(n-3) + a(n-4), with a(0)=1, a(1)=a(2)=0, a(3)=1.

1, 0, 0, 1, 1, 0, 1, 2, 1, 1, 3, 3, 2, 4, 6, 5, 6, 10, 11, 11, 16, 21, 22, 27, 37, 43, 49, 64, 80, 92, 113, 144, 172, 205, 257, 316, 377, 462, 573, 693, 839, 1035, 1266, 1532, 1874, 2301, 2798, 3406, 4175, 5099, 6204, 7581, 9274, 11303, 13785, 16855, 20577, 25088

Offset

0,8

Comments

Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=3, I={0,1}. - Vladimir Baltic, Mar 07 2012

Number of compositions (ordered partitions) of n into parts 3 and 4.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics 4 (2010), 119-135.

Johann Cigler, Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials, arXiv:2212.02118 [math.NT], 2022.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 484

Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.

E. Wilson, The Scales of Mt. Meru

Index entries for two-way infinite sequences

Index entries for linear recurrences with constant coefficients, signature (0,0,1,1).

Formula

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

a(n)/a(n-1) tends to A060007. - Gary W. Adamson, Oct 22 2006

a(n) = Sum_{k=0..floor(n/3)} binomial(k,n-3*k). - Seiichi Manyama, Mar 06 2019

Mathematica

LinearRecurrence[{0, 0, 1, 1}, {1, 0, 0, 1}, 60] (* G. C. Greubel, Mar 05 2019 *)

Prog

(PARI) a(n)=polcoeff(if(n<0, (1+x)/(1+x-x^4), 1/(1-x^3-x^4)) +x*O(x^abs(n)), abs(n))
(Magma) I:=[1, 0, 0, 1]; [n le 4 select I[n] else Self(n-3) +Self(n-4): n in [1..60]]; // G. C. Greubel, Mar 05 2019
(Sage) (1/(1-x^3-x^4)).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Mar 05 2019
(GAP) a:=[1, 0, 0, 1];; for n in [5..60] do a[n]:=a[n-3]+a[n-4]; od; a; # G. C. Greubel, Mar 05 2019

Crossrefs

A003269(n) = a(-4-n)(-1)^n.

Sequence in context: A346798 A340274 A127838 * A363567 A284834 A279677

Adjacent sequences: A017814 A017815 A017816 * A017818 A017819 A017820

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 17 1999

Status

approved