A003274 - OEIS

A003274

Number of key permutations of length n: permutations {a_i} with |a_i - a_{i-1}| = 1 or 2.
(Formerly M1583)

1, 1, 2, 6, 12, 20, 34, 56, 88, 136, 208, 314, 470, 700, 1038, 1534, 2262, 3330, 4896, 7192, 10558, 15492, 22724, 33324, 48860, 71630, 105002, 153912, 225594, 330650, 484618, 710270, 1040980, 1525660, 2235994, 3277040, 4802768, 7038832, 10315944, 15118786

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OFFSET

0,3

REFERENCES

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 251 terms from R. H. Hardin)

S. Avgustinovich and S. Kitaev, On uniquely k-determined permutations, Discr. Math., 308 (2008), 1500-1507.

Hugh Denoncourt, Ordinal pattern probabilities for symmetric random walks, arXiv:1907.07172 [math.CO], 2019.

E. S. Page, Systematic generation of ordered sequences using recurrence relations, Computer J., 14 (1971), 150-153.

E. S. Page, Systematic generation of ordered sequences using recurrence relations, The Computer Journal 14 (1971), 150-153. (Annotated scanned copy)

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.

Index entries for linear recurrences with constant coefficients, signature (3, -3, 2, -2, 1).

FORMULA

For n > 1, a(n) = 2*A069241(n).

G.f.: -(x^6 - x^5 + x^3 + 2*x^2 - 2*x + 1)/((x^3 + x - 1)*(x-1)^2).

Limit_{n->oo} a(n+1)/a(n) = A092526 = 1/A263719. - Alois P. Heinz, Apr 15 2018

MAPLE

A003274:=-(1-z+3*z**2-2*z**3+z**5)/(z**3+z-1)/(z-1)**2; # [Conjectured by Simon Plouffe in his 1992 dissertation.]

MATHEMATICA

CoefficientList[Series[-(x^6 - x^5 + x^3 + 2 x^2 - 2 x + 1)/((x^3 + x - 1) (x - 1)^2), {x, 0, 39}], x] (* Michael De Vlieger, Oct 01 2019 *)

CROSSREFS

Cf. A069241, A092526, A174700, A263719, A302118.