A274541 Decimal expansion of exp(sqrt(2)/2).

2, 0, 2, 8, 1, 1, 4, 9, 8, 1, 6, 4, 7, 4, 7, 2, 4, 5, 1, 1, 0, 8, 1, 2, 6, 1, 1, 2, 7, 4, 6, 3, 5, 1, 1, 7, 5, 1, 7, 4, 3, 2, 5, 0, 9, 2, 5, 4, 2, 6, 1, 3, 5, 2, 0, 6, 1, 7, 7, 7, 5, 9, 7, 2, 1, 2, 2, 2, 1, 5, 3, 9, 5, 0, 4, 8, 7, 1, 6, 5, 5, 9, 4, 2, 5, 9, 6

Offset

1,1

Comments

Define P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k), n >= 1 and P(0) = 1 with x(2) = (sqrt(2) + 1) and x(n) = 1 for all other n.

We find that C2 = lim_{n->infinity} P(n) = exp(sqrt(2)/2).

The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

The Dev Team and Simon Plouffe, The Inverse Symbolic Calculator (ISC).

Formula

c = exp(sqrt(2)/2).

c = lim_{n->infinity} P(n), with P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k), for n >= 1, and P(0) = 1, with x(2) = (1 + sqrt(2)), the silver mean A014176, and x(n) = 1 for all other n.

Example

c = 2.02811498164747245110812611274635117517432509254...

Maple

Digits := 140: evalf(exp(sqrt(2)/2)); # End program 1.
P := proc(n) : if n=0 then 1 else P(n) := expand((1/n)*(add(x(n-k)*P(k), k=0..n-1))) fi; end: x := proc(n): if n=2 then (sqrt(2)+1) else 1 fi: end:
Digits := 140: evalf(P(250)); # End program 2.

Mathematica

First@ RealDigits@ N[Exp[Sqrt[2]/2], 83] (* Michael De Vlieger, Jun 27 2016 *)

Prog

(PARI) my(x=exp(sqrt(2)/2)); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; print1(d, ", ")) \\ Felix Fröhlich, Jun 27 2016
(Magma) SetDefaultRealField(RealField(100)); Exp[Sqrt[2]/2]; // G. C. Greubel, Aug 19 2018

Crossrefs

Cf. A274540, A274542, A014176, A010503, A036039.

Sequence in context: A211888 A350601 A293783 * A301772 A021497 A201735

Adjacent sequences: A274538 A274539 A274540 * A274542 A274543 A274544

Keyword

cons,nonn

Author

Johannes W. Meijer, Jun 27 2016

Extensions

More digits from Jon E. Schoenfield, Mar 15 2018

Status

approved