A009949 Factorial expansion of sqrt(2) = Sum_{n>=1} a(n)/n!, using greedy algorithm.

1, 0, 2, 1, 4, 4, 1, 5, 0, 8, 1, 11, 1, 7, 8, 4, 4, 4, 11, 13, 1, 6, 15, 13, 8, 12, 22, 25, 14, 9, 13, 11, 30, 9, 16, 25, 3, 12, 11, 2, 35, 41, 29, 29, 11, 27, 43, 32, 1, 16, 2, 5, 29, 3, 2, 30, 18, 30, 32, 56, 44, 38, 44, 27, 4

Offset

1,3

Links

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

Index entries for factorial base representation

Example

sqrt(2) = 1 + 0/2! + 2/3! + 1/4! + 4/5! + 4/6! + 1/7! + 5/8! + ...

Maple

A009949 := proc(a, n) local i, b, c; b := a; c := [ floor(b) ]; for i from 1 to n-1 do b := b-c[ i ]/i!; c := [ op(c), floor(b*(i+1)!) ]; od; c; end:

Mathematica

With[{b = Sqrt[2]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Dec 10 2018 *)

Prog

(PARI) default(realprecision, 250); b = sqrt(2); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Dec 10 2018
(PARI) default(realprecision, 900); my(t=sqrt(2)); for(n=1, 80, t=t*n; print1(floor(t), ", "); t=frac(t)); \\ Joerg Arndt, Dec 17 2018
(Magma) SetDefaultRealField(RealField(250));  [Floor(Sqrt(2))] cat [Floor(Factorial(n)*Sqrt(2)) - n*Floor(Factorial((n-1))*Sqrt(2)) : n in [2..80]]; // G. C. Greubel, Dec 10 2018
(Sage) b=sqrt(2);
def a(n):
    if (n==1): return floor(b)
    else: return expand(floor(factorial(n)*b) - n*floor(factorial(n-1)*b))
[a(n) for n in (1..80)] # G. C. Greubel, Dec 10 2018

Crossrefs

Cf. A002193 (decimal expansion), A040000 (continued fraction).

Cf. A067881 (sqrt(3)), A068446 (sqrt(5)), A320839 (sqrt(7)).

Sequence in context: A274826 A129862 A137593 * A070785 A214985 A105537

Adjacent sequences: A009946 A009947 A009948 * A009950 A009951 A009952

Keyword

nonn

Author

N. J. A. Sloane, Bill Gosper

Status

approved