|
|
A163271
|
|
Numerators of fractions in a 'zero-transform' approximation of sqrt(2) by means of a(n) = (a(n-1) + c)/(a(n-1) + 1) with c=2 and a(1)=0.
|
|
10
|
|
|
0, 2, 4, 10, 24, 58, 140, 338, 816, 1970, 4756, 11482, 27720, 66922, 161564, 390050, 941664, 2273378, 5488420, 13250218, 31988856, 77227930, 186444716, 450117362, 1086679440, 2623476242, 6333631924, 15290740090, 36915112104, 89120964298, 215157040700
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Twice the Pell numbers; for denominators see A001333 (numerators of the approximation of sqrt(2) for a(1) = 1).
Number of weak orderings R on {1,...,n} that are weakly single-peaked w.r.t. the total ordering 1 < ... < n and for which {1,...,n} has exactly one minimal and one maximal element for the weak ordering R. - J. Devillet, Sep 28 2017
|
|
LINKS
|
|
|
FORMULA
|
G.f.: x + x^2/(2*G(0)-x) where G(k) = 1 - (k+1)/(1 - x/(x +(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 07 2012
|
|
MAPLE
|
A163271:=gfun:-rectoproc({a(n) = 2 * a(n-1) + a(n-2), a(1) = 0, a(2) = 2}, a(n), remember): map(A163271, [$1..100]); # Muniru A Asiru, Oct 08 2017
|
|
MATHEMATICA
|
CoefficientList[Series[2*t^2/(1-2*t - t^2), {t, 0, 50}], t] (* or *) LinearRecurrence[{2, 1}, {0, 2}, 50] (* G. C. Greubel, Dec 12 2016 *)
|
|
PROG
|
(Haskell)
a163271 = sum . a128966_row . (subtract 1)
(PARI) concat([0], Vec(2*t^2/(1-2*t - t^2) + O(t^50))) \\ G. C. Greubel, Dec 12 2016
(GAP)
a := [0, 2];; for n in [3..10^2] do a[n] := 2*a[n-1] + a[n-2]; od; A163271:=a; # Muniru A Asiru, Oct 08 2017
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|