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

0,1

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

See A130620 for another version.

Sequence in context: A046979 A354096 A270086 * A016481 A303818 A047815

Adjacent sequences: A141408 A141409 A141410 * A141412 A141413 A141414

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