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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)
18
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; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 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->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
Sequence in context: A291876 A277365 A184432 * A259470 A121315 A078878
KEYWORD
nonn,easy
AUTHOR
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 March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)