A080013 Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=2, r=2, I={0,1}.

1, 0, 0, 1, 1, 1, 1, 3, 3, 4, 6, 9, 12, 16, 24, 33, 46, 64, 91, 127, 177, 249, 349, 489, 684, 960, 1345, 1884, 2640, 3700, 5185, 7264, 10180, 14265, 19989, 28009, 39249, 54999, 77067, 107992, 151326, 212049, 297136, 416368, 583444, 817561, 1145622, 1605324, 2249491, 3152139, 4416993

Offset

0,8

Comments

Also the number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=2, r=2, I={0,-1}.

References

Allen, Michael A., and Kenneth Edwards. "Identities relating permanents of some classes of (0, 1) Toeplitz matrices to generalized Fibonacci numbers." The Fibonacci Quarterly 63.2 (2025): 163-177.

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..6829

Michael A. Allen and Kenneth Edwards, Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices, arXiv:2107.02589 [math.CO], 2021.

Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (2010), 119-135

Tomislav Došlić, Mate Puljiz, Stjepan Šebek, and Josip Žubrinić, On a variant of Flory model, arXiv:2210.12411 [math.CO], 2022.

P. L. Krapivsky and J. M. Luck, Jamming and metastability in one dimension: from the kinetically constrained Ising chain to the Riviera model, arXiv:2211.12815 [cond-mat.stat-mech], 2022.

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

Formula

Recurrence: a(n) = a(n-2)+a(n-3)+a(n-4)-a(n-6).

G.f.: -(x^2-1)/(x^6-x^4-x^3-x^2+1)

Mathematica

LinearRecurrence[{0, 1, 1, 1, 0, -1}, {1, 0, 0, 1, 1, 1}, 60] (* Harvey P. Dale, Aug 08 2019 *)

Prog

(PARI) a(n)=([0, 1, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0; 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 1; -1, 0, 1, 1, 1, 0]^n*[1; 0; 0; 1; 1; 1])[1, 1] \\ Charles R Greathouse IV, May 20 2026

Crossrefs

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014.

Sequence in context: A241036 A241050 A363237 * A396585 A152949 A338431

Adjacent sequences: A080010 A080011 A080012 * A080014 A080015 A080016

Keyword

nonn,easy

Author

Vladimir Baltic, Jan 24 2003

Status

approved