OFFSET
0,4
REFERENCES
Lehmer, D. H.; Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.
R. P. Stanley, Enumerative Combinatorics I, p. 252, Example 4.7.16.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties , arXiv:1103.2582 [math.CO], 2011-2013.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,-1).
FORMULA
G.f.: (1-x)/((1+x)*(1 - 2*x + x^2 - 2*x^3 + x^4)).
a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) - a(n-5).
a(n) = h(n) - h(n-1), n>0, h(n) = Sum_{k=1..n} (Sum_{r=0..k} (C(k,r)*Sum_{m=0..r}(C(r,m)*Sum_{j=0..m} C(m,j)*C(j,n-m-k-j-r)*(-1)^(n-m-k-j-r) ))). - Vladimir Kruchinin, Sep 10 2010
Limit_{n -> oo} a(n)/a(n-1) = (1 + sqrt(2) + sqrt(2*sqrt(2)-1)) /2 = 1.88320350591... for n>2. Limit_{n -> oo} a(n-1)/a(n) = (1 + sqrt(2) - sqrt(2*sqrt(2)-1)) /2 = 0.53101005645... for n>0. - Tim Monahan, Aug 09 2011
7*a(n) = 2*(-1)^n - 8*A112575(n) - 2*A112575(n-2) + 6*A112575(n-1) + 5*A112575(n+1). - R. J. Mathar, Sep 27 2013
Empirical: a(n) + a(n+1) = A183324(n). - R. J. Mathar, Sep 27 2013
MATHEMATICA
LinearRecurrence[{1, 1, 1, 1, -1}, {1, 0, 1, 2, 4}, 40] (* Harvey P. Dale, Aug 28 2012 *)
PROG
(Maxima) h(n) := sum(sum(binomial(k, r) *sum(binomial(r, m) *sum(binomial(m, j) *binomial(j, n-m-k-j-r) *(-1)^(n-m-k-j-r), j, 0, m), m, 0, r), r, 0, k), k, 1, n); a(n):=h(n)-h(n-1); /* Vladimir Kruchinin, Sep 10 2010 */
(Magma) I:=[1, 0, 1, 2, 4]; [n le 5 select I[n] else Self(n-1) +Self(n-2) +Self(n-3) +Self(n-4) -Self(n-5): n in [1..41]]; // G. C. Greubel, Jan 14 2022
(SageMath) [( (1-x)/((1+x)*(1-2*x+x^2-2*x^3+x^4)) ).series(x, n+1).list()[n] for n in (0..40)] # G. C. Greubel, Jan 14 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
New description from Max Alekseyev, Jul 09 2006
STATUS
approved