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A003274 Number of key permutations of length n: permutations {a_i} with |a_i-a_{i-1}| = 1 or 2.
(Formerly M1583)
17
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 (list; graph; refs; listen; history; text; internal format)
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.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

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->infinity} 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.

Sequence in context: A291876 A277365 A184432 * A259470 A121315 A078878

Adjacent sequences:  A003271 A003272 A003273 * A003275 A003276 A003277

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Better description and g.f. from Erich Friedman.

a(0)=1 prepended and g.f. adapted by Alois P. Heinz, Apr 01 2018

STATUS

approved

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Last modified June 2 14:21 EDT 2020. Contains 334787 sequences. (Running on oeis4.)