A007759 Knopfmacher expansion of sqrt(2): a(2n) = 2*(a(2n-1) + 1)^2 - 1, a(2n+1) = 2*(a(2n)^2 - 1).

2, 17, 576, 665857, 886731088896, 1572584048032918633353217, 4946041176255201878775086487573351061418968498176, 48926646634423881954586808839856694558492182258668537145547700898547222910968507268117381704646657

Offset: 1

Author

Arnold Knopfmacher

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..11
• A. Knopfmacher and J. Knopfmacher, An alternating product representation for real numbers, in Applications of Fibonacci numbers, Vol. 3 (Kluwer 1990), pp. 209-216.

Crossrefs

Cf. A002193 (sqrt(2)), A001601.

Programs

Magma

function a(n)
  if n eq 1 then return 2;
  elif n mod 2 eq 0 then return 2*(a(n-1) +1)^2 -1;
  else return 2*(a(n-1)^2 -1);
  end if; return a; end function;
[a(n): n in [1..9]]; // G. C. Greubel, Mar 04 2020

Maple

a:= proc(n) option remember;
if n=1 then 2
elif `mod`(n,2) = 0 then 2*(a(n-1) +1)^2 -1
else 2*(a(n-1)^2 -1)
end if; end proc;
seq(a(n), n = 1..9); # G. C. Greubel, Mar 04 2020

Mathematica

a[n_]:= a[n]= If[n==1, 2, If[EvenQ[n], 2*(a[n-1] +1)^2 -1, 2*a[n-1]^2 -2]]; Table[a[n], {n,9}] (* G. C. Greubel, Mar 04 2020 *)

PARI

a(n) = if (n==1, 2, if (n % 2, 2*a(n-1)^2 - 2, 2*(a(n-1)+1)^2 - 1)); \\ Michel Marcus, Feb 20 2019

Sage

@CachedFunction
def a(n):
    if (n==1): return 2
    elif (n%2==0): return 2*(a(n-1) +1)^2 -1
    else: return 2*(a(n-1)^2 -1)
[a(n) for n in (1..9)] # G. C. Greubel, Mar 04 2020

Extensions

More terms from Christian G. Bower, Jan 06 2006