A141411 Defined in comments.

3, 1, 31, 28, 365, 514, 4388, 8220, 53871, 122284, 673222, 1748055, 8535397, 24383499, 109449848, 334783855, 1415768769, 4548229589, 18434398665, 61345927764, 241210652738, 823296868656, 3167642169823, 11010462627756, 41708741708554, 146886286090602

Offset: 0

Author

Paul Curtz, Jun 18 2007

Keywords

• nonn
• easy

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

Crossrefs

See A130620 for another version.

Programs

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 *)

Formula

a(n) ~ c * d^n, where d = 3.6412947999106071671946396356753..., c = 1.387705266307957334035092183546... . - Vaclav Kotesovec, Sep 10 2014

Extensions

Edited by N. J. A. Sloane, Aug 26 2009

Corrected and extended by Alois P. Heinz, Sep 06 2009