A130620 Defined in comments.

3, 9, 31, 106, 365, 1263, 4388, 15336, 53871, 190059, 673222, 2393291, 8535397, 30526712, 109449848, 393272258, 1415768769, 5105086517, 18434398665, 66647658995, 241210652738, 873773659486, 3167642169823, 11491042716338, 41708741708554, 151461799255253

Offset: 0

Author

Paul Curtz, Jun 18 2007

Keywords

• nonn
• base
• 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 a(n) is the sum of the odd coefficients of P(n,x) if n is odd and a(n) is the sum of the even coefficients otherwise: a(n) = ((-1)^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).

Examples

			We have P(0,x)=3, P(1,x)=1+9x, P(2,x)=4+6x+27x^2, ..., so that for example a(2) = (25+37)/2 = 31.
The polynomials P(n,x) are:
n=0: 3,
n=1: 1+ 9*x,
n=2: 4+ 6*x+ 27*x^2,
n=3: 1+25*x+ 27*x^2+ 81*x^3,
n=4: 5+14*x+117*x^2+108*x^3+243*x^4,
n=5: 9+48*x+100*x^2+486*x^3+405*x^4+729*x^5.
		

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 A141411 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) +(-1)^n*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] + (-1)^n*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... (same as for A141411), c = 1.38770526630795733403509218... . - Vaclav Kotesovec, Sep 12 2014

Extensions

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

Definition corrected and more terms from Alois P. Heinz, Sep 06 2009