OFFSET
0,3
COMMENTS
a(n) is the number of perfect matchings in the circulant graph with 2*n vertices with jumps 1 and 3. - Robert Israel, Jan 24 2019
REFERENCES
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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..400
N. Metropolis, M. L. Stein, and P. R. Stein, Permanents of cyclic (0,1) matrices, Journal of Combinatorial Theory, Volume 7, Issue 4, December 1969, Pages 291-321.
Earl Glen Whitehead, Jr., Four-discordant permutations, J. Austral. Math. Soc. Ser. A 28 (1979), no. 3, 369-377.
Index entries for linear recurrences with constant coefficients, signature (2,0,0,-1).
FORMULA
G.f.: (1 - x + 2*x^3 + 13*x^4 - 3*x^5 - 6*x^6 - 10*x^7)/(1 - 2*x + x^4).
a(n) = 2*a(n-1) - a(n-4); a(0)=1, a(1)=1, a(2)=2, a(3)=6, a(4)=24, a(5)=44, a(6)=80, a(7)=144. - Harvey P. Dale, Dec 13 2011
MATHEMATICA
Join[{1, 1, 2, 6}, LinearRecurrence[{2, 0, 0, -1}, {24, 44, 80, 144}, 40]] (* or *) CoefficientList[ Series[ (1-x+2x^3+13x^4- 3x^5- 6x^6- 10x^7)/ (1-2x+ x^4), {x, 0, 40}], x] (* Harvey P. Dale, Dec 13 2011 *)
PROG
(PARI) my(x='x+O('x^40)); Vec((1-x+2*x^3+13*x^4-3*x^5-6*x^6-10*x^7)/(1 -2*x+x^4)) \\ G. C. Greubel, Apr 22 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+ 2*x^3+13*x^4-3*x^5-6*x^6-10*x^7)/(1-2*x+x^4) )); // G. C. Greubel, Apr 22 2019
(Sage) ((1-x+2*x^3+13*x^4-3*x^5-6*x^6-10*x^7)/(1-2*x+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 22 2019
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved