OFFSET
0,1
COMMENTS
Given any sequence {u(i), i >= 0} we define a family of polynomials by P(0,x) = u(0), P(n,x) = u(n) + x*Sum_{ i=0..n-1 } (u(i)*P(n-i-1, x). Then we set a(n) = (P(n,-1)+P(n,1))/2.
For the present example we take {u(i)} to be 3,1,4,1,5,9,... (A000796).
REFERENCES
P. Curtz, Gazette des Mathematiciens, 1992, 52, p.44.
P. Flajolet, X. Gourdon and B. Salvy, Gazette des Mathematiciens, 1993, 55, pp.67-78 .
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
FORMULA
a(n) ~ c * d^n, where d = 3.6412947999106071671946396356753..., c = 1.387705266307957334035092183546... . - Vaclav Kotesovec, Sep 10 2014
MAPLE
u:= proc(n) Digits:= max(n+10);
trunc (10* frac(evalf(Pi*10^(n-1))))
end:
P:= proc(n) option remember; local i, x;
if n=0 then u(0)
else unapply(expand(u(n)+x*add(u(i)*P(n-i-1)(x), i=0..n-1)), x)
fi
end:
a:= n-> (P(n)(1)+P(n)(-1))/2:
seq(a(n), n=0..30); # Alois P. Heinz, Sep 06 2009
MATHEMATICA
nmax = 25; digits = RealDigits[Pi, 10, nmax+1][[1]]; p[0][_] = digits[[1]]; p[n_][x_] := p[n][x] = digits[[n+1]] + x*Sum[digits[[i+1]] p[n-i-1][x], {i, 0, n-1}]; a[n_] := (p[n][1] + p[n][-1])/2; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Nov 22 2012 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Jun 18 2007
EXTENSIONS
Edited by N. J. A. Sloane, Aug 26 2009
Corrected and extended by Alois P. Heinz, Sep 06 2009
STATUS
approved